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Introduction to Confidence Intervals

Slide Duration:

Section 1: Introduction
Descriptive Statistics vs. Inferential Statistics

25m 31s

Intro
0:00
0:10
0:11
Statistics
0:35
Statistics
0:36
Let's Think About High School Science
1:12
Measurement and Find Patterns (Mathematical Formula)
1:13
Statistics = Math of Distributions
4:58
Distributions
4:59
Problematic… but also GREAT
5:58
Statistics
7:33
How is It Different from Other Specializations in Mathematics?
7:34
Statistics is Fundamental in Natural and Social Sciences
7:53
Two Skills of Statistics
8:20
Description (Exploration)
8:21
Inference
9:13
Descriptive Statistics vs. Inferential Statistics: Apply to Distributions
9:58
Descriptive Statistics
9:59
Inferential Statistics
11:05
Populations vs. Samples
12:19
Populations vs. Samples: Is it the Truth?
12:20
Populations vs. Samples: Pros & Cons
13:36
Populations vs. Samples: Descriptive Values
16:12
Putting Together Descriptive/Inferential Stats & Populations/Samples
17:10
Putting Together Descriptive/Inferential Stats & Populations/Samples
17:11
Example 1: Descriptive Statistics vs. Inferential Statistics
19:09
Example 2: Descriptive Statistics vs. Inferential Statistics
20:47
Example 3: Sample, Parameter, Population, and Statistic
21:40
Example 4: Sample, Parameter, Population, and Statistic
23:28
Section 2: About Samples: Cases, Variables, Measurements

32m 14s

Intro
0:00
Data
0:09
Data, Cases, Variables, and Values
0:10
Rows, Columns, and Cells
2:03
Example: Aircrafts
3:52
How Do We Get Data?
5:38
Research: Question and Hypothesis
5:39
Research Design
7:11
Measurement
7:29
Research Analysis
8:33
Research Conclusion
9:30
Types of Variables
10:03
Discrete Variables
10:04
Continuous Variables
12:07
Types of Measurements
14:17
Types of Measurements
14:18
Types of Measurements (Scales)
17:22
Nominal
17:23
Ordinal
19:11
Interval
21:33
Ratio
24:24
Example 1: Cases, Variables, Measurements
25:20
Example 2: Which Scale of Measurement is Used?
26:55
Example 3: What Kind of a Scale of Measurement is This?
27:26
Example 4: Discrete vs. Continuous Variables.
30:31
Section 3: Visualizing Distributions
Introduction to Excel

8m 9s

Intro
0:00
Before Visualizing Distribution
0:10
Excel
0:11
Excel: Organization
0:45
Workbook
0:46
Column x Rows
1:50
Tools: Menu Bar, Standard Toolbar, and Formula Bar
3:00
Excel + Data
6:07
Exce and Data
6:08
Frequency Distributions in Excel

39m 10s

Intro
0:00
0:08
Data in Excel and Frequency Distributions
0:09
Raw Data to Frequency Tables
0:42
Raw Data to Frequency Tables
0:43
Frequency Tables: Using Formulas and Pivot Tables
1:28
Example 1: Number of Births
7:17
Example 2: Age Distribution
20:41
Example 3: Height Distribution
27:45
Example 4: Height Distribution of Males
32:19
Frequency Distributions and Features

25m 29s

Intro
0:00
0:10
Data in Excel, Frequency Distributions, and Features of Frequency Distributions
0:11
Example #1
1:35
Uniform
1:36
Example #2
2:58
Unimodal, Skewed Right, and Asymmetric
2:59
Example #3
6:29
Bimodal
6:30
Example #4a
8:29
Symmetric, Unimodal, and Normal
8:30
Point of Inflection and Standard Deviation
11:13
Example #4b
12:43
Normal Distribution
12:44
Summary
13:56
Uniform, Skewed, Bimodal, and Normal
13:57
17:34
Sketch Problem 2: Life Expectancy
20:01
Sketch Problem 3: Telephone Numbers
22:01
Sketch Problem 4: Length of Time Used to Complete a Final Exam
23:43
Dotplots and Histograms in Excel

42m 42s

Intro
0:00
0:06
0:07
Previously
1:02
Data, Frequency Table, and visualization
1:03
Dotplots
1:22
Dotplots Excel Example
1:23
Dotplots: Pros and Cons
7:22
Pros and Cons of Dotplots
7:23
Dotplots Excel Example Cont.
9:07
Histograms
12:47
Histograms Overview
12:48
Example of Histograms
15:29
Histograms: Pros and Cons
31:39
Pros
31:40
Cons
32:31
Frequency vs. Relative Frequency
32:53
Frequency
32:54
Relative Frequency
33:36
Example 1: Dotplots vs. Histograms
34:36
Example 2: Age of Pennies Dotplot
36:21
Example 3: Histogram of Mammal Speeds
38:27
Example 4: Histogram of Life Expectancy
40:30
Stemplots

12m 23s

Intro
0:00
0:05
0:06
What Sets Stemplots Apart?
0:46
Data Sets, Dotplots, Histograms, and Stemplots
0:47
Example 1: What Do Stemplots Look Like?
1:58
Example 2: Back-to-Back Stemplots
5:00
7:46
Example 4: Quiz Grade & Afterschool Tutoring Stemplot
9:56
Bar Graphs

22m 49s

Intro
0:00
0:05
0:08
Review of Frequency Distributions
0:44
Y-axis and X-axis
0:45
Types of Frequency Visualizations Covered so Far
2:16
Introduction to Bar Graphs
4:07
Example 1: Bar Graph
5:32
Example 1: Bar Graph
5:33
Do Shapes, Center, and Spread of Distributions Apply to Bar Graphs?
11:07
Do Shapes, Center, and Spread of Distributions Apply to Bar Graphs?
11:08
Example 2: Create a Frequency Visualization for Gender
14:02
Example 3: Cases, Variables, and Frequency Visualization
16:34
Example 4: What Kind of Graphs are Shown Below?
19:29
Section 4: Summarizing Distributions
Central Tendency: Mean, Median, Mode

38m 50s

Intro
0:00
0:07
0:08
Central Tendency 1
0:56
Way to Summarize a Distribution of Scores
0:57
Mode
1:32
Median
2:02
Mean
2:36
Central Tendency 2
3:47
Mode
3:48
Median
4:20
Mean
5:25
Summation Symbol
6:11
Summation Symbol
6:12
Population vs. Sample
10:46
Population vs. Sample
10:47
Excel Examples
15:08
Finding Mode, Median, and Mean in Excel
15:09
Median vs. Mean
21:45
Effect of Outliers
21:46
Relationship Between Parameter and Statistic
22:44
Type of Measurements
24:00
Which Distributions to Use With
24:55
Example 1: Mean
25:30
Example 2: Using Summation Symbol
29:50
Example 3: Average Calorie Count
32:50
Example 4: Creating an Example Set
35:46
Variability

42m 40s

Intro
0:00
0:05
0:06
0:45
0:46
5:45
5:46
Range, Quartiles and Interquartile Range
6:37
Range
6:38
Interquartile Range
8:42
Interquartile Range Example
10:58
Interquartile Range Example
10:59
Variance and Standard Deviation
12:27
Deviations
12:28
Sum of Squares
14:35
Variance
16:55
Standard Deviation
17:44
Sum of Squares (SS)
18:34
Sum of Squares (SS)
18:35
Population vs. Sample SD
22:00
Population vs. Sample SD
22:01
Population vs. Sample
23:20
Mean
23:21
SD
23:51
Example 1: Find the Mean and Standard Deviation of the Variable Friends in the Excel File
27:21
Example 2: Find the Mean and Standard Deviation of the Tagged Photos in the Excel File
35:25
Example 3: Sum of Squares
38:58
Example 4: Standard Deviation
41:48
Five Number Summary & Boxplots

57m 15s

Intro
0:00
0:06
0:07
Summarizing Distributions
0:37
0:38
5 Number Summary
1:14
Boxplot: Visualizing 5 Number Summary
3:37
Boxplot: Visualizing 5 Number Summary
3:38
Boxplots on Excel
9:01
Using 'Stocks' and Using Stacked Columns
9:02
Boxplots on Excel Example
10:14
When are Boxplots Useful?
32:14
Pros
32:15
Cons
32:59
How to Determine Outlier Status
33:24
Rule of Thumb: Upper Limit
33:25
Rule of Thumb: Lower Limit
34:16
Signal Outliers in an Excel Data File Using Conditional Formatting
34:52
Modified Boxplot
48:38
Modified Boxplot
48:39
Example 1: Percentage Values & Lower and Upper Whisker
49:10
Example 2: Boxplot
50:10
Example 3: Estimating IQR From Boxplot
53:46
Example 4: Boxplot and Missing Whisker
54:35
Shape: Calculating Skewness & Kurtosis

41m 51s

Intro
0:00
0:16
0:17
Skewness Concept
1:09
Skewness Concept
1:10
Calculating Skewness
3:26
Calculating Skewness
3:27
Interpreting Skewness
7:36
Interpreting Skewness
7:37
Excel Example
8:49
Kurtosis Concept
20:29
Kurtosis Concept
20:30
Calculating Kurtosis
24:17
Calculating Kurtosis
24:18
Interpreting Kurtosis
29:01
Leptokurtic
29:35
Mesokurtic
30:10
Platykurtic
31:06
Excel Example
32:04
Example 1: Shape of Distribution
38:28
Example 2: Shape of Distribution
39:29
Example 3: Shape of Distribution
40:14
Example 4: Kurtosis
41:10
Normal Distribution

34m 33s

Intro
0:00
0:13
0:14
What is a Normal Distribution
0:44
The Normal Distribution As a Theoretical Model
0:45
Possible Range of Probabilities
3:05
Possible Range of Probabilities
3:06
What is a Normal Distribution
5:07
Can Be Described By
5:08
Properties
5:49
'Same' Shape: Illusion of Different Shape!
7:35
'Same' Shape: Illusion of Different Shape!
7:36
Types of Problems
13:45
Example: Distribution of SAT Scores
13:46
Shape Analogy
19:48
Shape Analogy
19:49
Example 1: The Standard Normal Distribution and Z-Scores
22:34
Example 2: The Standard Normal Distribution and Z-Scores
25:54
Example 3: Sketching and Normal Distribution
28:55
Example 4: Sketching and Normal Distribution
32:32
Standard Normal Distributions & Z-Scores

41m 44s

Intro
0:00
0:06
0:07
A Family of Distributions
0:28
Infinite Set of Distributions
0:29
Transforming Normal Distributions to 'Standard' Normal Distribution
1:04
Normal Distribution vs. Standard Normal Distribution
2:58
Normal Distribution vs. Standard Normal Distribution
2:59
Z-Score, Raw Score, Mean, & SD
4:08
Z-Score, Raw Score, Mean, & SD
4:09
Weird Z-Scores
9:40
Weird Z-Scores
9:41
Excel
16:45
For Normal Distributions
16:46
For Standard Normal Distributions
19:11
Excel Example
20:24
Types of Problems
25:18
Percentage Problem: P(x)
25:19
Raw Score and Z-Score Problems
26:28
Standard Deviation Problems
27:01
Shape Analogy
27:44
Shape Analogy
27:45
Example 1: Deaths Due to Heart Disease vs. Deaths Due to Cancer
28:24
Example 2: Heights of Male College Students
33:15
Example 3: Mean and Standard Deviation
37:14
Example 4: Finding Percentage of Values in a Standard Normal Distribution
37:49
Normal Distribution: PDF vs. CDF

55m 44s

Intro
0:00
0:15
0:16
Frequency vs. Cumulative Frequency
0:56
Frequency vs. Cumulative Frequency
0:57
Frequency vs. Cumulative Frequency
4:32
Frequency vs. Cumulative Frequency Cont.
4:33
Calculus in Brief
6:21
Derivative-Integral Continuum
6:22
PDF
10:08
PDF for Standard Normal Distribution
10:09
PDF for Normal Distribution
14:32
Integral of PDF = CDF
21:27
Integral of PDF = CDF
21:28
Example 1: Cumulative Frequency Graph
23:31
Example 2: Mean, Standard Deviation, and Probability
24:43
Example 3: Mean and Standard Deviation
35:50
Example 4: Age of Cars
49:32
Section 5: Linear Regression
Scatterplots

47m 19s

Intro
0:00
0:04
0:05
Previous Visualizations
0:30
Frequency Distributions
0:31
Compare & Contrast
2:26
Frequency Distributions Vs. Scatterplots
2:27
Summary Values
4:53
Shape
4:54
Center & Trend
6:41
8:22
Univariate & Bivariate
10:25
Example Scatterplot
10:48
Shape, Trend, and Strength
10:49
Positive and Negative Association
14:05
Positive and Negative Association
14:06
Linearity, Strength, and Consistency
18:30
Linearity
18:31
Strength
19:14
Consistency
20:40
Summarizing a Scatterplot
22:58
Summarizing a Scatterplot
22:59
Example 1: Gapminder.org, Income x Life Expectancy
26:32
Example 2: Gapminder.org, Income x Infant Mortality
36:12
Example 3: Trend and Strength of Variables
40:14
Example 4: Trend, Strength and Shape for Scatterplots
43:27
Regression

32m 2s

Intro
0:00
0:05
0:06
Linear Equations
0:34
Linear Equations: y = mx + b
0:35
Rough Line
5:16
Rough Line
5:17
Regression - A 'Center' Line
7:41
Reasons for Summarizing with a Regression Line
7:42
Predictor and Response Variable
10:04
Goal of Regression
12:29
Goal of Regression
12:30
Prediction
14:50
Example: Servings of Mile Per Year Shown By Age
14:51
Intrapolation
17:06
Extrapolation
17:58
Error in Prediction
20:34
Prediction Error
20:35
Residual
21:40
Example 1: Residual
23:34
Example 2: Large and Negative Residual
26:30
Example 3: Positive Residual
28:13
Example 4: Interpret Regression Line & Extrapolate
29:40
Least Squares Regression

56m 36s

Intro
0:00
0:13
0:14
Best Fit
0:47
Best Fit
0:48
Sum of Squared Errors (SSE)
1:50
Sum of Squared Errors (SSE)
1:51
Why Squared?
3:38
Why Squared?
3:39
Quantitative Properties of Regression Line
4:51
Quantitative Properties of Regression Line
4:52
So How do we Find Such a Line?
6:49
SSEs of Different Line Equations & Lowest SSE
6:50
Carl Gauss' Method
8:01
How Do We Find Slope (b1)
11:00
How Do We Find Slope (b1)
11:01
Hoe Do We Find Intercept
15:11
Hoe Do We Find Intercept
15:12
Example 1: Which of These Equations Fit the Above Data Best?
17:18
Example 2: Find the Regression Line for These Data Points and Interpret It
26:31
Example 3: Summarize the Scatterplot and Find the Regression Line.
34:31
Example 4: Examine the Mean of Residuals
43:52
Correlation

43m 58s

Intro
0:00
0:05
0:06
Summarizing a Scatterplot Quantitatively
0:47
Shape
0:48
Trend
1:11
Strength: Correlation ®
1:45
Correlation Coefficient ( r )
2:30
Correlation Coefficient ( r )
2:31
Trees vs. Forest
11:59
Trees vs. Forest
12:00
Calculating r
15:07
Average Product of z-scores for x and y
15:08
Relationship between Correlation and Slope
21:10
Relationship between Correlation and Slope
21:11
Example 1: Find the Correlation between Grams of Fat and Cost
24:11
Example 2: Relationship between r and b1
30:24
Example 3: Find the Regression Line
33:35
Example 4: Find the Correlation Coefficient for this Set of Data
37:37
Correlation: r vs. r-squared

52m 52s

Intro
0:00
0:07
0:08
R-squared
0:44
What is the Meaning of It? Why Squared?
0:45
Parsing Sum of Squared (Parsing Variability)
2:25
SST = SSR + SSE
2:26
What is SST and SSE?
7:46
What is SST and SSE?
7:47
r-squared
18:33
Coefficient of Determination
18:34
If the Correlation is Strong…
20:25
If the Correlation is Strong…
20:26
If the Correlation is Weak…
22:36
If the Correlation is Weak…
22:37
Example 1: Find r-squared for this Set of Data
23:56
Example 2: What Does it Mean that the Simple Linear Regression is a 'Model' of Variance?
33:54
Example 3: Why Does r-squared Only Range from 0 to 1
37:29
Example 4: Find the r-squared for This Set of Data
39:55
Transformations of Data

27m 8s

Intro
0:00
0:05
0:06
Why Transform?
0:26
Why Transform?
0:27
Shape-preserving vs. Shape-changing Transformations
5:14
Shape-preserving = Linear Transformations
5:15
Shape-changing Transformations = Non-linear Transformations
6:20
Common Shape-Preserving Transformations
7:08
Common Shape-Preserving Transformations
7:09
Common Shape-Changing Transformations
8:59
Powers
9:00
Logarithms
9:39
Change Just One Variable? Both?
10:38
Log-log Transformations
10:39
Log Transformations
14:38
Example 1: Create, Graph, and Transform the Data Set
15:19
Example 2: Create, Graph, and Transform the Data Set
20:08
Example 3: What Kind of Model would You Choose for this Data?
22:44
Example 4: Transformation of Data
25:46
Section 6: Collecting Data in an Experiment
Sampling & Bias

54m 44s

Intro
0:00
0:05
0:06
Descriptive vs. Inferential Statistics
1:04
Descriptive Statistics: Data Exploration
1:05
Example
2:03
To tackle Generalization…
4:31
Generalization
4:32
Sampling
6:06
'Good' Sample
6:40
Defining Samples and Populations
8:55
Population
8:56
Sample
11:16
Why Use Sampling?
13:09
Why Use Sampling?
13:10
Goal of Sampling: Avoiding Bias
15:04
What is Bias?
15:05
Where does Bias Come from: Sampling Bias
17:53
Where does Bias Come from: Response Bias
18:27
Sampling Bias: Bias from Bas Sampling Methods
19:34
Size Bias
19:35
Voluntary Response Bias
21:13
Convenience Sample
22:22
Judgment Sample
23:58
25:40
Response Bias: Bias from 'Bad' Data Collection Methods
28:00
Nonresponse Bias
29:31
Questionnaire Bias
31:10
Incorrect Response or Measurement Bias
37:32
Example 1: What Kind of Biases?
40:29
Example 2: What Biases Might Arise?
44:46
Example 3: What Kind of Biases?
48:34
Example 4: What Kind of Biases?
51:43
Sampling Methods

14m 25s

Intro
0:00
0:05
0:06
Biased vs. Unbiased Sampling Methods
0:32
Biased Sampling
0:33
Unbiased Sampling
1:13
Probability Sampling Methods
2:31
Simple Random
2:54
Stratified Random Sampling
4:06
Cluster Sampling
5:24
Two-staged Sampling
6:22
Systematic Sampling
7:25
8:33
Example 2: Describe How to Take a Two-Stage Sample from this Book
10:16
Example 3: Sampling Methods
11:58
Example 4: Cluster Sample Plan
12:48
Research Design

53m 54s

Intro
0:00
0:06
0:07
Descriptive vs. Inferential Statistics
0:51
Descriptive Statistics: Data Exploration
0:52
Inferential Statistics
1:02
Variables and Relationships
1:44
Variables
1:45
Relationships
2:49
Not Every Type of Study is an Experiment…
4:16
Category I - Descriptive Study
4:54
Category II - Correlational Study
5:50
Category III - Experimental, Quasi-experimental, Non-experimental
6:33
Category III
7:42
Experimental, Quasi-experimental, and Non-experimental
7:43
Why CAN'T the Other Strategies Determine Causation?
10:18
Third-variable Problem
10:19
Directionality Problem
15:49
What Makes Experiments Special?
17:54
Manipulation
17:55
Control (and Comparison)
21:58
Methods of Control
26:38
Holding Constant
26:39
Matching
29:11
Random Assignment
31:48
Experiment Terminology
34:09
'true' Experiment vs. Study
34:10
Independent Variable (IV)
35:16
Dependent Variable (DV)
35:45
Factors
36:07
Treatment Conditions
36:23
Levels
37:43
Confounds or Extraneous Variables
38:04
Blind
38:38
Blind Experiments
38:39
Double-blind Experiments
39:29
How Categories Relate to Statistics
41:35
Category I - Descriptive Study
41:36
Category II - Correlational Study
42:05
Category III - Experimental, Quasi-experimental, Non-experimental
42:43
Example 1: Research Design
43:50
Example 2: Research Design
47:37
Example 3: Research Design
50:12
Example 4: Research Design
52:00
Between and Within Treatment Variability

41m 31s

Intro
0:00
0:06
0:07
Experimental Designs
0:51
Experimental Designs: Manipulation & Control
0:52
Two Types of Variability
2:09
Between Treatment Variability
2:10
Within Treatment Variability
3:31
Updated Goal of Experimental Design
5:47
Updated Goal of Experimental Design
5:48
Example: Drugs and Driving
6:56
Example: Drugs and Driving
6:57
Different Types of Random Assignment
11:27
All Experiments
11:28
Completely Random Design
12:02
Randomized Block Design
13:19
Randomized Block Design
15:48
Matched Pairs Design
15:49
Repeated Measures Design
19:47
Between-subject Variable vs. Within-subject Variable
22:43
Completely Randomized Design
22:44
Repeated Measures Design
25:03
Example 1: Design a Completely Random, Matched Pair, and Repeated Measures Experiment
26:16
Example 2: Block Design
31:41
Example 3: Completely Randomized Designs
35:11
Example 4: Completely Random, Matched Pairs, or Repeated Measures Experiments?
39:01
Section 7: Review of Probability Axioms
Sample Spaces

37m 52s

Intro
0:00
0:07
0:08
Why is Probability Involved in Statistics
0:48
Probability
0:49
Can People Tell the Difference between Cheap and Gourmet Coffee?
2:08
Taste Test with Coffee Drinkers
3:37
If No One can Actually Taste the Difference
3:38
If Everyone can Actually Taste the Difference
5:36
Creating a Probability Model
7:09
Creating a Probability Model
7:10
D'Alembert vs. Necker
9:41
D'Alembert vs. Necker
9:42
Problem with D'Alembert's Model
13:29
Problem with D'Alembert's Model
13:30
Covering Entire Sample Space
15:08
Fundamental Principle of Counting
15:09
Where Do Probabilities Come From?
22:54
Observed Data, Symmetry, and Subjective Estimates
22:55
Checking whether Model Matches Real World
24:27
Law of Large Numbers
24:28
Example 1: Law of Large Numbers
27:46
Example 2: Possible Outcomes
30:43
Example 3: Brands of Coffee and Taste
33:25
Example 4: How Many Different Treatments are there?
35:33

20m 29s

Intro
0:00
0:08
0:09
Disjoint Events
0:41
Disjoint Events
0:42
Meaning of 'or'
2:39
In Regular Life
2:40
In Math/Statistics/Computer Science
3:10
3:55
If A and B are Disjoint: P (A and B)
3:56
If A and B are Disjoint: P (A or B)
5:15
5:41
5:42
8:31
If A and B are not Disjoint: P (A or B)
8:32
Example 1: Which of These are Mutually Exclusive?
10:50
Example 2: What is the Probability that You will Have a Combination of One Heads and Two Tails?
12:57
Example 3: Engagement Party
15:17
Example 4: Home Owner's Insurance
18:30
Conditional Probability

57m 19s

Intro
0:00
0:05
0:06
'or' vs. 'and' vs. Conditional Probability
1:07
'or' vs. 'and' vs. Conditional Probability
1:08
'and' vs. Conditional Probability
5:57
P (M or L)
5:58
P (M and L)
8:41
P (M|L)
11:04
P (L|M)
12:24
Tree Diagram
15:02
Tree Diagram
15:03
Defining Conditional Probability
22:42
Defining Conditional Probability
22:43
Common Contexts for Conditional Probability
30:56
Medical Testing: Positive Predictive Value
30:57
Medical Testing: Sensitivity
33:03
Statistical Tests
34:27
Example 1: Drug and Disease
36:41
Example 2: Marbles and Conditional Probability
40:04
Example 3: Cards and Conditional Probability
45:59
Example 4: Votes and Conditional Probability
50:21
Independent Events

24m 27s

Intro
0:00
0:05
0:06
Independent Events & Conditional Probability
0:26
Non-independent Events
0:27
Independent Events
2:00
Non-independent and Independent Events
3:08
Non-independent and Independent Events
3:09
Defining Independent Events
5:52
Defining Independent Events
5:53
Multiplication Rule
7:29
Previously…
7:30
But with Independent Evens
8:53
Example 1: Which of These Pairs of Events are Independent?
11:12
Example 2: Health Insurance and Probability
15:12
Example 3: Independent Events
17:42
Example 4: Independent Events
20:03
Section 8: Probability Distributions
Introduction to Probability Distributions

56m 45s

Intro
0:00
0:08
0:09
Sampling vs. Probability
0:57
Sampling
0:58
Missing
1:30
What is Missing?
3:06
Insight: Probability Distributions
5:26
Insight: Probability Distributions
5:27
What is a Probability Distribution?
7:29
From Sample Spaces to Probability Distributions
8:44
Sample Space
8:45
Probability Distribution of the Sum of Two Die
11:16
The Random Variable
17:43
The Random Variable
17:44
Expected Value
21:52
Expected Value
21:53
Example 1: Probability Distributions
28:45
Example 2: Probability Distributions
35:30
Example 3: Probability Distributions
43:37
Example 4: Probability Distributions
47:20
Expected Value & Variance of Probability Distributions

53m 41s

Intro
0:00
0:06
0:07
Discrete vs. Continuous Random Variables
1:04
Discrete vs. Continuous Random Variables
1:05
Mean and Variance Review
4:44
Mean: Sample, Population, and Probability Distribution
4:45
Variance: Sample, Population, and Probability Distribution
9:12
Example Situation
14:10
Example Situation
14:11
Some Special Cases…
16:13
Some Special Cases…
16:14
Linear Transformations
19:22
Linear Transformations
19:23
What Happens to Mean and Variance of the Probability Distribution?
20:12
n Independent Values of X
25:38
n Independent Values of X
25:39
Compare These Two Situations
30:56
Compare These Two Situations
30:57
Two Random Variables, X and Y
32:02
Two Random Variables, X and Y
32:03
Example 1: Expected Value & Variance of Probability Distributions
35:35
Example 2: Expected Values & Standard Deviation
44:17
Example 3: Expected Winnings and Standard Deviation
48:18
Binomial Distribution

55m 15s

Intro
0:00
0:05
0:06
Discrete Probability Distributions
1:42
Discrete Probability Distributions
1:43
Binomial Distribution
2:36
Binomial Distribution
2:37
Multiplicative Rule Review
6:54
Multiplicative Rule Review
6:55
How Many Outcomes with k 'Successes'
10:23
Adults and Bachelor's Degree: Manual List of Outcomes
10:24
P (X=k)
19:37
Putting Together # of Outcomes with the Multiplicative Rule
19:38
Expected Value and Standard Deviation in a Binomial Distribution
25:22
Expected Value and Standard Deviation in a Binomial Distribution
25:23
Example 1: Coin Toss
33:42
38:03
Example 3: Types of Blood and Probability
45:39
Example 4: Expected Number and Standard Deviation
51:11
Section 9: Sampling Distributions of Statistics
Introduction to Sampling Distributions

48m 17s

Intro
0:00
0:08
0:09
Probability Distributions vs. Sampling Distributions
0:55
Probability Distributions vs. Sampling Distributions
0:56
Same Logic
3:55
Logic of Probability Distribution
3:56
Example: Rolling Two Die
6:56
Simulating Samples
9:53
To Come Up with Probability Distributions
9:54
In Sampling Distributions
11:12
Connecting Sampling and Research Methods with Sampling Distributions
12:11
Connecting Sampling and Research Methods with Sampling Distributions
12:12
Simulating a Sampling Distribution
14:14
Experimental Design: Regular Sleep vs. Less Sleep
14:15
Logic of Sampling Distributions
23:08
Logic of Sampling Distributions
23:09
General Method of Simulating Sampling Distributions
25:38
General Method of Simulating Sampling Distributions
25:39
Questions that Remain
28:45
Questions that Remain
28:46
Example 1: Mean and Standard Error of Sampling Distribution
30:57
Example 2: What is the Best Way to Describe Sampling Distributions?
37:12
Example 3: Matching Sampling Distributions
38:21
Example 4: Mean and Standard Error of Sampling Distribution
41:51
Sampling Distribution of the Mean

1h 8m 48s

Intro
0:00
0:05
0:06
Special Case of General Method for Simulating a Sampling Distribution
1:53
Special Case of General Method for Simulating a Sampling Distribution
1:54
Computer Simulation
3:43
Using Simulations to See Principles behind Shape of SDoM
15:50
Using Simulations to See Principles behind Shape of SDoM
15:51
Conditions
17:38
Using Simulations to See Principles behind Center (Mean) of SDoM
20:15
Using Simulations to See Principles behind Center (Mean) of SDoM
20:16
Conditions: Does n Matter?
21:31
Conditions: Does Number of Simulation Matter?
24:37
Using Simulations to See Principles behind Standard Deviation of SDoM
27:13
Using Simulations to See Principles behind Standard Deviation of SDoM
27:14
Conditions: Does n Matter?
34:45
Conditions: Does Number of Simulation Matter?
36:24
Central Limit Theorem
37:13
SHAPE
38:08
CENTER
39:34
39:52
Comparing Population, Sample, and SDoM
43:10
Comparing Population, Sample, and SDoM
43:11
48:24
What Happens When We Don't Know What the Population Looks Like?
48:25
Can We Have Sampling Distributions for Summary Statistics Other than the Mean?
49:42
How Do We Know whether a Sample is Sufficiently Unlikely?
53:36
Do We Always Have to Simulate a Large Number of Samples in Order to get a Sampling Distribution?
54:40
Example 1: Mean Batting Average
55:25
Example 2: Mean Sampling Distribution and Standard Error
59:07
Example 3: Sampling Distribution of the Mean
1:01:04
Sampling Distribution of Sample Proportions

54m 37s

Intro
0:00
0:06
0:07
Intro to Sampling Distribution of Sample Proportions (SDoSP)
0:51
Categorical Data (Examples)
0:52
Wish to Estimate Proportion of Population from Sample…
2:00
Notation
3:34
Population Proportion and Sample Proportion Notations
3:35
What's the Difference?
9:19
SDoM vs. SDoSP: Type of Data
9:20
SDoM vs. SDoSP: Shape
11:24
SDoM vs. SDoSP: Center
12:30
15:34
Binomial Distribution vs. Sampling Distribution of Sample Proportions
19:14
Binomial Distribution vs. SDoSP: Type of Data
19:17
Binomial Distribution vs. SDoSP: Shape
21:07
Binomial Distribution vs. SDoSP: Center
21:43
24:08
Example 1: Sampling Distribution of Sample Proportions
26:07
Example 2: Sampling Distribution of Sample Proportions
37:58
Example 3: Sampling Distribution of Sample Proportions
44:42
Example 4: Sampling Distribution of Sample Proportions
45:57
Section 10: Inferential Statistics
Introduction to Confidence Intervals

42m 53s

Intro
0:00
0:06
0:07
Inferential Statistics
0:50
Inferential Statistics
0:51
Two Problems with This Picture…
3:20
Two Problems with This Picture…
3:21
Solution: Confidence Intervals (CI)
4:59
Solution: Hypotheiss Testing (HT)
5:49
Which Parameters are Known?
6:45
Which Parameters are Known?
6:46
Confidence Interval - Goal
7:56
When We Don't Know m but know s
7:57
When We Don't Know
18:27
When We Don't Know m nor s
18:28
Example 1: Confidence Intervals
26:18
Example 2: Confidence Intervals
29:46
Example 3: Confidence Intervals
32:18
Example 4: Confidence Intervals
38:31
t Distributions

1h 2m 6s

Intro
0:00
0:04
0:05
When to Use z vs. t?
1:07
When to Use z vs. t?
1:08
What is z and t?
3:02
z-score and t-score: Commonality
3:03
z-score and t-score: Formulas
3:34
z-score and t-score: Difference
5:22
Why not z? (Why t?)
7:24
Why not z? (Why t?)
7:25
But Don't Worry!
15:13
Gossett and t-distributions
15:14
Rules of t Distributions
17:05
t-distributions are More Normal as n Gets Bigger
17:06
t-distributions are a Family of Distributions
18:55
Degrees of Freedom (df)
20:02
Degrees of Freedom (df)
20:03
t Family of Distributions
24:07
t Family of Distributions : df = 2 , 4, and 60
24:08
df = 60
29:16
df = 2
29:59
How to Find It?
31:01
'Student's t-distribution' or 't-distribution'
31:02
Excel Example
33:06
Example 1: Which Distribution Do You Use? Z or t?
45:26
47:41
Example 3: t Distributions
52:15
Example 4: t Distributions , confidence interval, and mean
55:59
Introduction to Hypothesis Testing

1h 6m 33s

Intro
0:00
0:06
0:07
Issues to Overcome in Inferential Statistics
1:35
Issues to Overcome in Inferential Statistics
1:36
What Happens When We Don't Know What the Population Looks Like?
2:57
How Do We Know whether a sample is Sufficiently Unlikely
3:43
Hypothesizing a Population
6:44
Hypothesizing a Population
6:45
Null Hypothesis
8:07
Alternative Hypothesis
8:56
Hypotheses
11:58
Hypotheses
11:59
Errors in Hypothesis Testing
14:22
Errors in Hypothesis Testing
14:23
Steps of Hypothesis Testing
21:15
Steps of Hypothesis Testing
21:16
Single Sample HT ( When Sigma Available)
26:08
26:09
Step1
27:08
Step 2
27:58
Step 3
28:17
Step 4
32:18
Single Sample HT (When Sigma Not Available)
36:33
36:34
Step1: Hypothesis Testing
36:58
Step 2: Significance Level
37:25
Step 3: Decision Stage
37:40
Step 4: Sample
41:36
Sigma and p-value
45:04
Sigma and p-value
45:05
On tailed vs. Two Tailed Hypotheses
45:51
Example 1: Hypothesis Testing
48:37
Example 2: Heights of Women in the US
57:43
Example 3: Select the Best Way to Complete This Sentence
1:03:23
Confidence Intervals for the Difference of Two Independent Means

55m 14s

Intro
0:00
0:14
0:15
One Mean vs. Two Means
1:17
One Mean vs. Two Means
1:18
Notation
2:41
A Sample! A Set!
2:42
Mean of X, Mean of Y, and Difference of Two Means
3:56
SE of X
4:34
SE of Y
6:28
Sampling Distribution of the Difference between Two Means (SDoD)
7:48
Sampling Distribution of the Difference between Two Means (SDoD)
7:49
Rules of the SDoD (similar to CLT!)
15:00
Mean for the SDoD Null Hypothesis
15:01
Standard Error
17:39
When can We Construct a CI for the Difference between Two Means?
21:28
Three Conditions
21:29
Finding CI
23:56
One Mean CI
23:57
Two Means CI
25:45
Finding t
29:16
Finding t
29:17
Interpreting CI
30:25
Interpreting CI
30:26
Better Estimate of s (s pool)
34:15
Better Estimate of s (s pool)
34:16
Example 1: Confidence Intervals
42:32
Example 2: SE of the Difference
52:36
Hypothesis Testing for the Difference of Two Independent Means

50m

Intro
0:00
0:06
0:07
The Goal of Hypothesis Testing
0:56
One Sample and Two Samples
0:57
Sampling Distribution of the Difference between Two Means (SDoD)
3:42
Sampling Distribution of the Difference between Two Means (SDoD)
3:43
Rules of the SDoD (Similar to CLT!)
6:46
Shape
6:47
Mean for the Null Hypothesis
7:26
Standard Error for Independent Samples (When Variance is Homogenous)
8:18
Standard Error for Independent Samples (When Variance is not Homogenous)
9:25
Same Conditions for HT as for CI
10:08
Three Conditions
10:09
Steps of Hypothesis Testing
11:04
Steps of Hypothesis Testing
11:05
Formulas that Go with Steps of Hypothesis Testing
13:21
Step 1
13:25
Step 2
14:18
Step 3
15:00
Step 4
16:57
Example 1: Hypothesis Testing for the Difference of Two Independent Means
18:47
Example 2: Hypothesis Testing for the Difference of Two Independent Means
33:55
Example 3: Hypothesis Testing for the Difference of Two Independent Means
44:22
Confidence Intervals & Hypothesis Testing for the Difference of Two Paired Means

1h 14m 11s

Intro
0:00
0:09
0:10
The Goal of Hypothesis Testing
1:27
One Sample and Two Samples
1:28
Independent Samples vs. Paired Samples
3:16
Independent Samples vs. Paired Samples
3:17
Which is Which?
5:20
Independent SAMPLES vs. Independent VARIABLES
7:43
independent SAMPLES vs. Independent VARIABLES
7:44
T-tests Always…
10:48
T-tests Always…
10:49
Notation for Paired Samples
12:59
Notation for Paired Samples
13:00
Steps of Hypothesis Testing for Paired Samples
16:13
Steps of Hypothesis Testing for Paired Samples
16:14
Rules of the SDoD (Adding on Paired Samples)
18:03
Shape
18:04
Mean for the Null Hypothesis
18:31
Standard Error for Independent Samples (When Variance is Homogenous)
19:25
Standard Error for Paired Samples
20:39
Formulas that go with Steps of Hypothesis Testing
22:59
Formulas that go with Steps of Hypothesis Testing
23:00
Confidence Intervals for Paired Samples
30:32
Confidence Intervals for Paired Samples
30:33
Example 1: Confidence Intervals & Hypothesis Testing for the Difference of Two Paired Means
32:28
Example 2: Confidence Intervals & Hypothesis Testing for the Difference of Two Paired Means
44:02
Example 3: Confidence Intervals & Hypothesis Testing for the Difference of Two Paired Means
52:23
Type I and Type II Errors

31m 27s

Intro
0:00
0:18
0:19
Errors and Relationship to HT and the Sample Statistic?
1:11
Errors and Relationship to HT and the Sample Statistic?
1:12
7:00
One Sample t-test: Friends on Facebook
7:01
Two Sample t-test: Friends on Facebook
13:46
Usually, Lots of Overlap between Null and Alternative Distributions
16:59
Overlap between Null and Alternative Distributions
17:00
How Distributions and 'Box' Fit Together
22:45
How Distributions and 'Box' Fit Together
22:46
Example 1: Types of Errors
25:54
Example 2: Types of Errors
27:30
Example 3: What is the Danger of the Type I Error?
29:38
Effect Size & Power

44m 41s

Intro
0:00
0:05
0:06
Distance between Distributions: Sample t
0:49
Distance between Distributions: Sample t
0:50
Problem with Distance in Terms of Standard Error
2:56
Problem with Distance in Terms of Standard Error
2:57
Test Statistic (t) vs. Effect Size (d or g)
4:38
Test Statistic (t) vs. Effect Size (d or g)
4:39
Rules of Effect Size
6:09
Rules of Effect Size
6:10
Why Do We Need Effect Size?
8:21
Tells You the Practical Significance
8:22
HT can be Deceiving…
10:25
Important Note
10:42
What is Power?
11:20
What is Power?
11:21
Why Do We Need Power?
14:19
Conditional Probability and Power
14:20
Power is:
16:27
Can We Calculate Power?
19:00
Can We Calculate Power?
19:01
How Does Alpha Affect Power?
20:36
How Does Alpha Affect Power?
20:37
How Does Effect Size Affect Power?
25:38
How Does Effect Size Affect Power?
25:39
How Does Variability and Sample Size Affect Power?
27:56
How Does Variability and Sample Size Affect Power?
27:57
How Do We Increase Power?
32:47
Increasing Power
32:48
Example 1: Effect Size & Power
35:40
Example 2: Effect Size & Power
37:38
Example 3: Effect Size & Power
40:55
Section 11: Analysis of Variance
F-distributions

24m 46s

Intro
0:00
0:04
0:05
Z- & T-statistic and Their Distribution
0:34
Z- & T-statistic and Their Distribution
0:35
F-statistic
4:55
The F Ration ( the Variance Ratio)
4:56
F-distribution
12:29
F-distribution
12:30
s and p-value
15:00
s and p-value
15:01
Example 1: Why Does F-distribution Stop At 0 But Go On Until Infinity?
18:33
Example 2: F-distributions
19:29
Example 3: F-distributions and Heights
21:29
ANOVA with Independent Samples

1h 9m 25s

Intro
0:00
0:05
0:06
The Limitations of t-tests
1:12
The Limitations of t-tests
1:13
Two Major Limitations of Many t-tests
3:26
Two Major Limitations of Many t-tests
3:27
Ronald Fisher's Solution… F-test! New Null Hypothesis
4:43
Ronald Fisher's Solution… F-test! New Null Hypothesis (Omnibus Test - One Test to Rule Them All!)
4:44
Analysis of Variance (ANoVA) Notation
7:47
Analysis of Variance (ANoVA) Notation
7:48
Partitioning (Analyzing) Variance
9:58
Total Variance
9:59
Within-group Variation
14:00
Between-group Variation
16:22
Time out: Review Variance & SS
17:05
Time out: Review Variance & SS
17:06
F-statistic
19:22
The F Ratio (the Variance Ratio)
19:23
S²bet = SSbet / dfbet
22:13
What is This?
22:14
How Many Means?
23:20
So What is the dfbet?
23:38
So What is SSbet?
24:15
S²w = SSw / dfw
26:05
What is This?
26:06
How Many Means?
27:20
So What is the dfw?
27:36
So What is SSw?
28:18
Chart of Independent Samples ANOVA
29:25
Chart of Independent Samples ANOVA
29:26
Example 1: Who Uploads More Photos: Unknown Ethnicity, Latino, Asian, Black, or White Facebook Users?
35:52
Hypotheses
35:53
Significance Level
39:40
Decision Stage
40:05
Calculate Samples' Statistic and p-Value
44:10
Reject or Fail to Reject H0
55:54
Example 2: ANOVA with Independent Samples
58:21
Repeated Measures ANOVA

1h 15m 13s

Intro
0:00
0:05
0:06
The Limitations of t-tests
0:36
Who Uploads more Pictures and Which Photo-Type is Most Frequently Used on Facebook?
0:37
ANOVA (F-test) to the Rescue!
5:49
Omnibus Hypothesis
5:50
Analyze Variance
7:27
Independent Samples vs. Repeated Measures
9:12
Same Start
9:13
Independent Samples ANOVA
10:43
Repeated Measures ANOVA
12:00
Independent Samples ANOVA
16:00
Same Start: All the Variance Around Grand Mean
16:01
Independent Samples
16:23
Repeated Measures ANOVA
18:18
Same Start: All the Variance Around Grand Mean
18:19
Repeated Measures
18:33
Repeated Measures F-statistic
21:22
The F Ratio (The Variance Ratio)
21:23
S²bet = SSbet / dfbet
23:07
What is This?
23:08
How Many Means?
23:39
So What is the dfbet?
23:54
So What is SSbet?
24:32
S² resid = SS resid / df resid
25:46
What is This?
25:47
So What is SS resid?
26:44
So What is the df resid?
27:36
SS subj and df subj
28:11
What is This?
28:12
How Many Subject Means?
29:43
So What is df subj?
30:01
So What is SS subj?
30:09
SS total and df total
31:42
What is This?
31:43
What is the Total Number of Data Points?
32:02
So What is df total?
32:34
so What is SS total?
32:47
Chart of Repeated Measures ANOVA
33:19
Chart of Repeated Measures ANOVA: F and Between-samples Variability
33:20
Chart of Repeated Measures ANOVA: Total Variability, Within-subject (case) Variability, Residual Variability
35:50
Example 1: Which is More Prevalent on Facebook: Tagged, Uploaded, Mobile, or Profile Photos?
40:25
Hypotheses
40:26
Significance Level
41:46
Decision Stage
42:09
Calculate Samples' Statistic and p-Value
46:18
Reject or Fail to Reject H0
57:55
Example 2: Repeated Measures ANOVA
58:57
Example 3: What's the Problem with a Bunch of Tiny t-tests?
1:13:59
Section 12: Chi-square Test
Chi-Square Goodness-of-Fit Test

58m 23s

Intro
0:00
0:05
0:06
Where Does the Chi-Square Test Belong?
0:50
Where Does the Chi-Square Test Belong?
0:51
A New Twist on HT: Goodness-of-Fit
7:23
HT in General
7:24
Goodness-of-Fit HT
8:26
12:17
Null Hypothesis
12:18
Alternative Hypothesis
13:23
Example
14:38
Chi-Square Statistic
17:52
Chi-Square Statistic
17:53
Chi-Square Distributions
24:31
Chi-Square Distributions
24:32
Conditions for Chi-Square
28:58
Condition 1
28:59
Condition 2
30:20
Condition 3
30:32
Condition 4
31:47
Example 1: Chi-Square Goodness-of-Fit Test
32:23
Example 2: Chi-Square Goodness-of-Fit Test
44:34
Example 3: Which of These Statements Describe Properties of the Chi-Square Goodness-of-Fit Test?
56:06
Chi-Square Test of Homogeneity

51m 36s

Intro
0:00
0:09
0:10
Goodness-of-Fit vs. Homogeneity
1:13
Goodness-of-Fit HT
1:14
Homogeneity
2:00
Analogy
2:38
5:00
Null Hypothesis
5:01
Alternative Hypothesis
6:11
Example
6:33
Chi-Square Statistic
10:12
Same as Goodness-of-Fit Test
10:13
Set Up Data
12:28
Setting Up Data Example
12:29
Expected Frequency
16:53
Expected Frequency
16:54
Chi-Square Distributions & df
19:26
Chi-Square Distributions & df
19:27
Conditions for Test of Homogeneity
20:54
Condition 1
20:55
Condition 2
21:39
Condition 3
22:05
Condition 4
22:23
Example 1: Chi-Square Test of Homogeneity
22:52
Example 2: Chi-Square Test of Homogeneity
32:10
Section 13: Overview of Statistics
Overview of Statistics

18m 11s

Intro
0:00
0:07
0:08
The Statistical Tests (HT) We've Covered
0:28
The Statistical Tests (HT) We've Covered
0:29
Organizing the Tests We've Covered…
1:08
One Sample: Continuous DV and Categorical DV
1:09
Two Samples: Continuous DV and Categorical DV
5:41
More Than Two Samples: Continuous DV and Categorical DV
8:21
The Following Data: OK Cupid
10:10
The Following Data: OK Cupid
10:11
Example 1: Weird-MySpace-Angle Profile Photo
10:38
Example 2: Geniuses
12:30
Example 3: Promiscuous iPhone Users
13:37
Example 4: Women, Aging, and Messaging
16:07

• ## Related Books

 0 answersPost by Michelle Greene on October 15, 2013Again, you are using Excel when we cannot use Excel on exams. Please show us with at scientific calculator... excel is not helpful but the calculator is very helpful. 0 answersPost by Brijesh Bolar on August 20, 2012What book are you referring to for these sessions.. Or what book do we refer.

### Introduction to Confidence Intervals

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

• Intro 0:00
• Inferential Statistics 0:50
• Inferential Statistics
• Two Problems with This Picture… 3:20
• Two Problems with This Picture…
• Solution: Confidence Intervals (CI)
• Solution: Hypotheiss Testing (HT)
• Which Parameters are Known? 6:45
• Which Parameters are Known?
• Confidence Interval - Goal 7:56
• When We Don't Know m but know s
• When We Don't Know 18:27
• When We Don't Know m nor s
• Example 1: Confidence Intervals 26:18
• Example 2: Confidence Intervals 29:46
• Example 3: Confidence Intervals 32:18
• Example 4: Confidence Intervals 38:31

### Transcription: Introduction to Confidence Intervals

Hi and welcome to www.educator.com.0000

Today we are going to be introduced to competence intervals.0002

Here is the roadmap for today, first we are going to do a brief overview of inferential statistics.0005

We have been trying to do some inferential statistics but there have been a couple of problems we keep running into.0013

So far I have fudged it.0022

We will address some of those problems head on and come up with 2 solutions.0024

One of those solutions is the competence interval and we are going to talk about competence intervals0031

when the sigma, population, standard deviation is known and when sigma is unknown.0039

Those are the two situations we are going to be focused on.0046

Let us go over inferential statistics.0049

We know the big picture idea there is some population represented by X and we wish we could know the population but we do not.0055

But instead what we can know is little samples.0065

We could know that but the problem is samples are biased.0071

Whenever we have samples and we summarize them using these mathematical summaries we call them statistics.0074

Just to give you an example of some statistics there things like x bar or s, those are all statistics.0084

What we would like to do is use these samples to understand something about the population.0093

Statistics, the field is about using these statistics to estimate parameters and0100

to give you ideas about parameters there are things like mu or sigma.0108

That is our whole goal.0112

Here we realize in order to jump from things like x bar and s to mu and sigma we are going to need more than just wishful thinking.0114

And that is where the sampling distributions come in.0132

Here we talk about sampling distribution often we are talking about some sort of statistic.0135

When we talk about sampling distribution of the mean we are talking about a whole bunch of x bars.0142

Here we have a whole bunch of x.0148

Here we have a whole bunch of x bar and that is the distribution.0150

When we summarize these statistics in the sampling distribution we call them expected values.0155

So it is not just mu, it is mu sub x bar.0164

It is not just sigma it is sigma sub x bar.0168

What we want to do is go from this to understand this but what we have learned0172

so far is how to see the relationship between parameters and expected values.0177

We know that these things have a relationship to each other.0186

And from doing that we could then make this jump.0189

It is like we use this to say something like this.0195

There are two problems with this picture although it seems rosy and there is still to nagging questions.0199

We would look at them a little bit before but we need to solve this more rigorously than we had before.0210

One question is this, what happens when we do not know what the population looks like?0217

Of course we could use the central limit theorem when we know mu and sigma from the population.0222

What if we do not know mu?0229

What if we do not know sigma?0231

Then what happens?0233

Also, how do we know whether a sample is sufficiently unlikely because remember the whole point0234

of the sampling distribution is for us to take sampling distributions from a known population and compare it to an unknown population.0240

If this sample does not match the sampling distribution enough that it is very unlikely to come from the sampling distribution.0254

We could say this is probably not the population that the sample came from.0261

How do we know when it sufficiently weird?0266

To answer these two questions there is going to be to solutions.0269

You can think of it as this one.0275

This first question roughly, they are both actually are answered in each of these but this one goes along better with that one.0281

This one goes along better with that one.0287

The two solutions are these, one is competence interval.0291

When we talk about competence interval here is what we are doing, we are going to figure out where mu might be from the sample.0302

We are going to try to figure out the population mu from the sample and0306

that is what we do when we do not know what the population looks like.0336

We try to figure it out from the sample.0342

Hypothesis testing actually takes another view.0344

The hypothesis testing, we come up with a hypothesis for what the population is like.0349

Hypothesize a population mu first.0355

In this case we are saying we are going to pull from something and figure out and pick a potential population mu.0363

And then we are going to test how weird the sample is.0376

We are going to come up with a number to tell us this is how weird the sample is.0387

We are going to decide is that weirdness weird enough?0393

That is going to be hypothesis testing.0398

But we are going to focus here on competence intervals.0401

Okay, when we talk about competence intervals we need to get an inventory of what we know so far.0404

Basically that is asking the question, which parameters are known or given to us?0413

What happens when we do not know what the population looks like?0418

Well we may not know what0422

The population looks like because we do not know anything about the population, or we know0424

Only a little bit about the population.0428

This is the case where we know a little.0431

Here we do not know mu but we do know sigma.0434

For some reason we have some partial information and that helps us out.0444

Here we know nothing.0450

Here nothing is helping us that we do not know mu and we are trying to figure it out but we do not know sigma either.0454

It is like nothing is helping us out here.0464

We just have to pull ourselves up from our own bootstraps.0466

These are the two situations that we are going to talk about it.0471

Here is the goal of competence interval.0475

The basic idea of the competence interval is going to be this.0480

We are going to try to figure out where mu might be but we do know x bar.0484

We know everything about the sample but we do not know anything about the population.0497

But in this case I am going to show you what happens when we already know sigma.0503

So we have a leg up.0508

We know sigma life is little that easier for us today.0509

Here is the thing, we do not know what the population looks like so cannot draw a normal or skewed or anything.0513

We have no idea what the population looks like and we have no idea what the population mu is.0524

But we for some reason know sigma is.0531

Sigma is given to us.0533

From there can we construct an SDOM?0534

Given that n is sufficiently large we can assume that it is normal.0540

We have no idea what mu is and so we do not know what mu sub x bar is.0548

We do not know it at all but we can figure out sigma sub x bar.0553

We could figure out the standard error because we have sigma and we could divide that by √n.0559

We have a little bit of information about the SDOM.0566

Here is what we do in competence intervals.0570

First assume that the x bar is the mu sub x bar.0574

Whatever your sample x bar is we are going to put back here.0586

We are going to assume it.0591

Here is why, because we always assume one thing to figure out the other,0595

here we are going to assume things about the x bar to figure out mu.0601

And hypothesis testing, we assume something about the population to figure out how0605

Weird x bar is.0609

Here because we know that the SDOM tends to be normal given a sufficiently0612

Large n what we know is that we can find out with reasonable competence what some0621

Significant borders are.0632

For instance, let us say we are one standard deviation away.0634

This is raw score and this is z score so we know at one standard deviation away0642

this base right here we know that that is 68% of SDOM.0650

Let us think about what this might mean.0660

When we get these borders what we might end up saying is that these are the borders in which 68% of our values will fall in the SDOM.0663

And here is what we could say we could also say that there is 68% chance that our0679

Population mu will fall in that zone.0686

That is a 68% competence interval.0691

For 68% is higher than half, but it is not that high.0697

But here is the thing we can have a high competence interval.0702

We can have a 95% competence interval or we can have a 99% competence interval.0707

That is what we can do.0713

We can have here is my x bar, here is 0 but what we can do is figure out0716

These borders such that we are now sure that 95% chance of having our0730

Population mean fall in this interval.0744

We can know that.0748

That is called the competence interval.0750

That is pretty hypothetically and you can even go to 99%.0753

And we could easily figure out these borders.0756

Here is how.0759

Because we easily figure out the border we could figure out what the z scores are.0761

This is what we call a two-tailed competence interval because even though the middle part is 95% that does not mean that part of 5%.0772

You will have 105% so that means that part is .025 so 2.5% and this part is .025.0785

And those the only parts that we are not sure.0796

There is a small chance that the population mean will fall somewhere out here but it is a very small chance.0798

We are trying to reduce it as much is possible.0809

Let us think about how we could find the z score out here.0812

We could use our tables in the back of the book, our z tables and we can look up and usually z tables will give you like one side.0817

We can look up .025 and look at the z score or we could do it on our Excel.0830

Instead of using normsdist, normsdist will give you the proportion of the distribution.0837

We are going to put in normsin as the inverse and here we want to put in the probability.0848

Now this is going to be my probability.0855

I am going to put in this probability .025 and we get 1.967.0870

This value here is -1.96 and because the normal distribution is symmetric we know that this part is also 1.960884

but now a positive instead of negative.0892

We know our z values on the end and if we know the z values what is our raw score here?0896

Tell me what this value is and also tell me what that value is.0908

Well the z score tells you how many standard errors away you are.0915

How many jumps away and each jump is worth that much.0921

We are away 1.96 of these jumps.0926

We are going to multiply this by this and then0931

Either subtract it from x or add it to x.0934

Step two in finding competence interval is let us say you want to find a 95% competence interval finds the z scores.0938

It is all in the case where you know sigma.0953

Step 3 is this, now you want to actually find the actual scores and that is going to be x bar + or -the z score × standard error.0957

That is what you are going to do.0984

And we know what the standard error is.0986

I am going to rewrite this to be x bar + or - z score × sigma / √n.0989

When we do that we could find these competence intervals.1003

Once you have these competence intervals then you that with 95% competence that1009

your population mean will fall in this interval between these two numbers.1019

Now the 95% is actually called the capture rate that is like 95% and 99%, whatever.1028

What would the competence interval be for 100%?1042

It would go from –infinity to infinity because that is how far the normal distribution goes.1047

But the capture rate is this the proportion of random sample for which this interval captures mu.1053

Let us imagine taking a whole bunch of random sample, it is going to be that 95% of the1080

Time those random samples in tail mu.1091

They somehow overlap with mu.1097

That is what we mean by 95% capture rate.1099

That is when you know sigma but now we do not know sigma.1103

We are in trouble but we do not know mu.1113

We do not sigma either.1115

Still our goal remains the same, we try to figure out mu from x bar.1116

But now we are a little hobbled.1128

I do not have a tool that I use to have.1132

The beginning part of the story stays the same.1135

The population we have no idea and from there we want to find the SDOM because1139

we are going to figure out how good our sample is.1146

We know the shape of our SDOM as long as our s is sufficiently big.1151

Can we figure out sigma sub x bar anymore?1157

No we cannot because we do not have sigma so how can figure out sigma sub x bar.1161

We cannot figure out that standard error.1170

Here is where another idea comes in.1171

There is another way we can estimate the standard error of the sampling distribution that is going to be s sub x bar.1175

Because we are going to use the sample standard deviation s instead of sigma.1186

Remember s is more variable, not quite right and because of that we corrected already a little bit by using n -1 instead of n.1200

Here we are going to divide that by √n.1214

If you double click on this you would see the square root of the sum of squares ÷ √ n -1.1218

You would see this inside of that.1231

We already tried to correct it a little bit, but s is still variable.1234

It is not quite as good as having sigma.1242

And there can be other problems that we run into.1245

This is pretty good though and it is a pretty good estimate but you always have1249

to keep in mind we have not as good of a standard error as we used to.1254

We have to account for that.1262

But the steps remain the same.1265

First assume x bar for mu sub x bar.1267

Two, find z for your capture rate.1275

If your capture rate for example 95% then you would find the z scores.1287

It is helpful to memorize that for this capture rate the z scores are going to be + or -1.96.1297

It is going to come up a lot.1305

Find the z scores for your capture rate.1306

Here we run into a problem.1310

I wish we could use z scores but here is an issue, we actually cannot because s is to variable for us to assume perfect normality.1314

And because of that we cannot use the z and instead we have to use the t which is very similar to z.1330

Find the t score for your capture rate.1348

Instead of having raw score and z score we are going to find t score.1352

For now you just need to know that you can find your t score in the back of the book but in1366

The next lesson we are going to go over why you use t and why you cannot use z.1372

That is a big story.1377

You are going to find t.1380

Once you find the t for your capture rate and that will also be + or -, t is going to be very similar to z score.1383

We are going to use this formula.1390

You are going to use a very similar idea to the z score competence interval where you want to know x bar + or -.1396

How a t score is also going to tell you how many standard errors away.1407

T × standard error.1411

But remember, you use t when you estimate this from sample.1417

If we unpack this, this is what it can look like x bar + or - t × this is that estimated standard error s/√n.1426

It is still the same idea.1443

It is how many jumps away, figuring that out and then multiplying that to the length of the jump1446

and adding that to x bar for the high-value and then subtracting that from the x bar for the low value.1451

In order to find t here is what you need to know for now.1458

You need to know whether it is a 1 or 2 tailed distribution.1465

If your competence interval is two-tailed then remember these are .0251470

because you would split the remaining 5% on both side.1478

But sometimes where t values though only give you one side.1482

They might give you a one sided 5% or one sided .25%.1487

You have to just keep in mind whether it is one tailed or two tailed and also the t distributions are a whole bunch of different distributions.1493

They are a whole bunch of different tables basically.1502

You have to also know what degrees of freedom.1508

For now you could remember degrees of freedom as n -1.1514

There are reasons for all of these things why we use t, why we use degrees of freedom all that stuff.1521

That will be covered in the next lesson.1528

For now, here is what you need to know.1529

You need to know whether it is one tailed or two tailed.1532

You also need to know degrees of freedom.1534

Once you have that you could actually look it up in t table usually found in the back of your book.1536

It might also be called the students t distribution because - invented it but he was actually contracted to work for Guinness.1542

That is why I cannot publish it under his actual name.1553

We published it under the pseudonym student because that is called the students t.1556

You can look up your degrees of freedom and then look for the area that you need and then go down and find the t score.1560

Very similar to z score.1573

Let us go on to some examples.1574

Example 1, consider two extreme situations n=10 and n=1,000.1582

If you use s in the formula for CI given sigma, here is the actual formula for when you have sigma.1591

We use 1.96 because we use the z score.1609

Which of these situations would you expect to give a capture rate closer to 95%?1614

Here is what this question is really asking.1621

When you know sigma for competence interval for 95% competence interval 1.96 that is my z × sigma / √n.1624

What it is asking you is what if you substituted in s?1649

Here we do not know sigma but we are going to just take this formula and use the z value s/√n.1656

In order to answer this question you really only need to keep in mind one thing, when is s more like sigma.1676

S is more like sigma when n is very large.1687

This situation would give you a very close capture rate of 95%.1708

This would be very, very similar.1721

However, when n is 10 you have more uncertainty and because of that the t distribution it is not as tight.1724

It is actually more like spread out and because of that, when n=10 you do not capture 95% just by being about 2 standard deviations out this way.1733

That would not capture 95% of those samples.1748

In fact you have to go out further to capture 95%.1753

This is going to be much closer to 95% capture rate.1758

This is going to give you a smaller capture rate.1763

That is because your s is going to be more variable and because of that your t distribution1766

is going to be more disperse because more variable means sort of wider.1778

95% CI for a population mean is calculated for random sample of weights and the resulting CI is from 42 to 48 pounds.1785

For each statement indicate whether it is a true or false interpretation of the CI.1798

This question is asking you do you understand what the competence interval means?1807

Do you understand what it is for?1811

Let us see, 95% of the weights in the population are between 42 and 48.1813

Does competence interval tell us about the actual population numbers?1821

No, it only tells us about the population mean.1830

This is actually not true.1833

We do not know anything about the actual numbers of the population.1836

We do not know whether it is skewed, whether it is uniform distribution.1840

We do not know any of those things.1847

The 95% thing would only be reasonable if the population was normal and its mu was exactly equal to x bar.1848

That would be the case.1862

That is not true.1864

95% of weights in the sample are between 42 and 48, does the CI tell us anything about this sample?1868

No, using the sample to estimate population mean.1878

We are using the SDOM.1882

We do not know anything about the sample itself.1884

That is also not true.1888

The probability that the interval includes the population mean is 95%.1893

This is actually true.1899

There is only a 5% chance that this interval does not contain the population mean.1902

The sample mean might not be in the competence interval.1919

That does not make sense if you look at the picture because we use the sample mean in order to construct the competence interval.1924

Of course this is in the competence intervals and this is just ridiculous.1932

Example 3, a random sample of 22 men had a mean body temperature of 98.1°, standard deviation of .73.1936

Construct a 95% competence interval for the mean of the population that the sample was drawn from.1950

Interpret the CI and 98.6° included in this.1956

This the average human body temperature.1963

We have body temperatures in the world and we do not know what that population looks like.1965

We are asking can we construct 95% competence interval such that whatever1975

the population mean is there is a 95% chance that we have covered it.1989

We start by assuming that the mean of the sample x bar is the mean of the sampling distribution of the mean.1994

We have done step one.2004

Step two is we have to construct CI and so here they give us x, but do we have sigma?2008

No.2023

We know that we cannot use the z score.2025

We have to use the t score.2029

Let us find the t for this.2031

This is .025 chance that we would not find it on the site and here is .025 chance that we can find it on the site.2033

What is the t scores?2043

This is the raw score or the temperature.2046

What is the t score for .025 when the degrees of freedom and that is n -1 there is 22 man so 22-1= 21 degrees of freedom. S2049

If you look in your book, at your students t distributions I am going to go down to where the df=21.2065

I am going to go across to where it says you know .025.2074

My table actually gives me this area so I am going to look at .025 on the side.2080

You and it says 2.08 is my t score.2086

That makes sense.2093

That is around 1.96.2095

You will see that as degrees of freedom get greater and greater this value becomes more and more close to 1.96.2098

On this side we know that it is symmetrical so I know it is -2.08.2108

From here I can construct my CI.2114

The CI is going to be the x bar + or – the t value × my standard error.2118

My estimated standard error here is s sub x bar because we do not have sigma.2129

That is going to be s ÷ √n.2137

Let us put in numbers here, so that is 98.1 that is our sample mean ± t value 2.08 × s .73 ÷ √22.2141

I am just going to calculate this on a calculator so that is going to be 98.1 and I will do the + side first. +2.08.2167

Excel does order of operation.2182

It needs to do the multiplication before the addition and its .3 ÷ √22.2185

That is the high-end of my competence interval is 98.4 and the low end is going to be 97.8.2195

98.4 and 97.8 are my CI.2217

When we interpret the competence interval we want to say something like2229

there is a 95% chance that the mean of the population lies between these two values.2239

Or another way we could say it is that if we draw samples at random, 95% of those samples will include the population mean.2250

95% of the samples in between this interval will include the population mean.2264

Is 98.6° included that is supposed to be the mean for everybody.2280

We see that it is not actually.2286

Maybe this sample is odd because our competence interval does not actually include the mean2288

that we secretly know for providing temperature of people.2297

That is when competence intervals are helpful.2307

Here is example 4, in a random sample of 1000 community college students, their mean score on a quantitative literacy test was 310.2310

The standard deviation on this test of all the community college students have taken is 360.2324

Construct a 95% competence interval for the mean of all community college students have ever taken this test.2331

Here is our random sample and their mean or x bar is 310 but the standard deviation2338

of all the students who have taken this test that is the sigma is 360.2351

Construct a 95% competence interval.2358

Well, the first part that we know population we do not know but we are given the population standard deviation.2361

And from that, let us construct the SDOM.2374

Well given that this n is quite large let us assume normality.2377

Here we could find out the standard error by putting 360 ÷ √ 1000.2382

Now going to our steps of our competence interval first we assume that x bar is the mean of our sampling distribution of the mean.2395

Here we could use the z instead of t because we have sigma and because of that we know that this is normal.2412

That is going to be +1.96 and -1.96 in order to construct a 95% competence interval.2425

Our CI is going to look something like this x bar + or – z × standard error.2436

If you sort of double click on standard error what you will find is sigma / √n.2446

Let us put in numbers here.2464

310 is our x bar.2467

Our z score is 1.96.2471

Our sigma is 360.2475

Our n is 1,000.2479

Let us put these in our calculators.2483

I will do the high end first 310 + 1.96 × 360 ÷√1,000.2487

Order of operations says it does not matter anything you multiply or divide it in.2508

That is my high end 332 as the high scoring end.2516

The low scoring end, the lower bound of my 95% CI is 287.7.2524

That is going to be 287.7 as well as 332.3.2537

The mean of the population 95% should fall between this interval.2547

That is the end for our competence intervals.2558

That is part one of competence intervals.2561

Hope you join me for t distributions to find out why we use t instead of z sometimes.2566

Thank you for using www.educator.com.2571

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