For more information, please see full course syllabus of Statistics

For more information, please see full course syllabus of Statistics

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### Hypothesis Testing for the Difference of Two Independent Means

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
- Roadmap
- The Goal of Hypothesis Testing
- Sampling Distribution of the Difference between Two Means (SDoD)
- Rules of the SDoD (Similar to CLT!)
- Shape
- Mean for the Null Hypothesis
- Standard Error for Independent Samples (When Variance is Homogenous)
- Standard Error for Independent Samples (When Variance is not Homogenous)
- Same Conditions for HT as for CI
- Steps of Hypothesis Testing
- Formulas that Go with Steps of Hypothesis Testing
- Example 1: Hypothesis Testing for the Difference of Two Independent Means
- Example 2: Hypothesis Testing for the Difference of Two Independent Means
- Example 3: Hypothesis Testing for the Difference of Two Independent Means

- Intro 0:00
- Roadmap 0:06
- Roadmap
- The Goal of Hypothesis Testing 0:56
- One Sample and Two Samples
- Sampling Distribution of the Difference between Two Means (SDoD) 3:42
- Sampling Distribution of the Difference between Two Means (SDoD)
- Rules of the SDoD (Similar to CLT!) 6:46
- Shape
- Mean for the Null Hypothesis
- Standard Error for Independent Samples (When Variance is Homogenous)
- Standard Error for Independent Samples (When Variance is not Homogenous)
- Same Conditions for HT as for CI 10:08
- Three Conditions
- Steps of Hypothesis Testing 11:04
- Steps of Hypothesis Testing
- Formulas that Go with Steps of Hypothesis Testing 13:21
- Step 1
- Step 2
- Step 3
- Step 4
- 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

### General Statistics Online Course

### Transcription: Hypothesis Testing for the Difference of Two Independent Means

*Hi and welcome to www.educator.com.*0000

*We are going to be talking about hypothesis testing for the difference between two independent means.*0001

*We are going to go over the goal of hypothesis testing in general.*0005

*We have only looked at it for one means so far, but we are going to look at*0012

*how it changes just very suddenly when we talk about two means.*0015

*We are going to re-talk about the sampling distribution of the difference between two means.*0019

*You have just watched the confidence interval for two means, then you do not need to watch this one.*0025

*You do not need to watch that section.*0032

*We are going to talk about the same conditions for doing hypothesis testing as first confidence interval.*0034

*They need to meet three conditions before you could do either of these two.*0043

*When we talk about the modified steps of hypothesis testing for two means and the formulas that go with those steps.*0047

*Let us talk about the goal of hypothesis testing.*0055

*In one sample what we wanted to do was reject the null if*0060

*we got a sample that was significantly different from the hypothesized mu.*0065

*For instance, significantly lower or significantly higher.*0073

*A significant does not mean important like it does in our modern use of the word.*0076

*It actually means does it standout?*0083

*Is it weird enough?*0086

*Does it stand out from the hypothesized mu?*0088

*In those cases we reject the null.*0091

*Our goal is to reject the null.*0095

*We can only say whether something is sufficiently weird w cannot say whether it is sufficiently similar.*0097

*Experiment is actually a success if they reject the null.*0106

*If they do not reject the null it is considered a null experiment or what we think of as uninformative which is not actually true.*0110

*That is how traditionally is that.*0118

*This is the case where we only have one sample and we have a hypothesized population.*0123

*Here we have two samples and in order to reject the null we need to get samples that are significantly different from each other.*0130

*They stand out from each other so x is different from y, y is different from x.*0144

*That is what we are really looking for.*0151

*Once again, just like the one sample, we cannot say whether they are sufficiently similar,*0154

*but we can say whether they are sufficiently different.*0159

*It is okay if x is significantly lower than y or significantly higher.*0163

*We do not really care.*0170

*We just care about significantly different.*0171

*If you do not care about which direction these are called two-tailed hypotheses.*0173

*Let us think if x and y are different from each other then x - y should not be 0.*0179

*But if x and y are exactly the same, x = y then x – y =0.*0189

*Because you can think about this as x – x because x – y.*0196

*If you want to think about it algebraically even if you add y to each side you would get perfectly x= y.*0201

*If x and y were the same, we should expect their difference to be 0.*0211

*Let us just review very briefly the sampling distribution of the difference between two means.*0218

*This is the case where we do not know what the population is like,*0228

*but because of the CLT we actually end up knowing quite a bit about the SDOM.*0233

*This is x the population of x and population of y.*0242

*This is the SDOM of x bar, so the whole bunch of x bars and this is the SDOM for y which is a whole bunch of y bars.*0247

*We know some things about these guys and we also know we can figure out the standard error from the sample.*0258

*What is nice about this if we do not need to know anything about the population.*0280

*All we have to do is know the standard deviation of the sample which we could easily calculate*0284

*in order to estimate the standard error of these two populations.*0288

*Once we have that now we can start talking about the SDOD (the sampling distribution of the difference between means).*0294

*What we want to do is instead of finding mu sub x or mu sub y, we want to know mu sub x bar – y bar.*0306

*Here you have to think of pulling out one sample from here and one sample from here getting the difference and plotting it.*0322

*If these guys are normal, we can assume this one to be normal.*0332

*Not only that but we can figure out the standard error of this guy as well just*0336

*from knowing these because the standard error is going to be square roots of s sub x ^{2}.*0342

*The variance of s/n sub x + variance of y/ n sub y.*0357

*These are all things that we have.*0366

*We do not need anything special.*0368

*We do not need sigma or anything like that.*0370

*We just need samples in order to calculate this.*0372

*If these two distributions or if these two distributions, the population distribution,*0374

*if we have a reason to suspect that these have homogeneous variance.*0384

*If their variances are the same then instead of s sub s ^{2} and s sub y^{2},*0389

*we can actually use spull ^{2} but we would not be doing that in this lesson, but you can.*0395

*Remember the rules of the SDOD are very similar to the CLT and if the SDOM for x is normal*0405

*and SDOM for y is normal then SDOD is normal too.*0415

*There is two ways that this could be true.*0419

*The first way is if populations are normal.*0421

*If population of x and y are normal then we could assume SDOM for x and y are normal.*0428

*Or are your other possibility is if n is large enough.*0435

*We want to talk about the mean for the null hypothesis.*0443

*The null hypotheses is saying that the population of x and population of y,*0450

*the difference between them is going to be 0 because they are similar.*0457

*The null hypotheses is saying both are similar, which means that the means of*0461

*the sampling distribution of the means, the SDOM means is going to be similar.*0467

*Which means that is strap in and will give us 0.*0474

*The null hypothesis says the mean of these differences of means it is going to be 0.*0478

*That is the null hypotheses and that is really saying that the SDOM for x and SDOM for the y are very similar.*0486

*Let us talk about standard error for independent samples.*0497

*Remember, we are still talking just about independent samples.*0502

*When variance is homogenous that is only used as Spull idea.*0506

*That means that x sub x bar - y bar is going to be equal to and pretend you are*0511

*writing just the regular idea where you are dividing by n sub x and n sub y.*0521

*Instead of using the variance from x and the variance from y, we are going to use that pulled variance idea.*0529

*That is going to be s pulled.*0536

*Some people think why do we just put that on top and put n sub x and n sub y at the bottom?*0547

*That will be algebraically wrong because remember, these are the denominators we would have*0554

*to have common denominators in order for us to put these together and we do not have common denominators yet.*0559

*What about in the case where variance is not homogenous and this is the vast majority of time and when in doubt,*0565

*when you do not know anything about the variance of the population go with this one.*0576

*It is just a safer option.*0582

*This is going to mean that this standard error is represented by the variance of x /n + variance of y /n.*0584

*Add these together and square the whole thing.*0602

*Just to recap, same conditions must be met in order to do hypothesis testing*0605

*for two means as the conditions for doing a confidence interval for two means.*0616

*It is that the two samples were randomly and independently selected from two different populations,*0622

*it is reasonable to assume that both populations that the sample come from are*0632

*normally distributed or the sample sizes are sufficiently large.*0636

*This was to ensure the normality of the SDOM.*0641

*Also in the case of the sample surveys, the population size should be at least 10 times larger than the sample size for each sample.*0643

*That is just assume so that we could assume replacement because probability actions change when you do not assume replacement.*0651

*Let us go in the steps of the hypothesis testing.*0663

*These are the same steps as you did when you have one mean, except now that we are subtly changing a few things.*0669

*I'm going to highlight those changes as we go through this.*0677

*First we need to state our hypotheses and remember now instead of having just the hypotheses that*0679

*the mean of the population equals this, what we are saying is that the mean of x,*0686

*population of x and the mean of the population of y those are the same.*0696

*Mu sub x - y will be 0.*0701

*You can also write it as mu sub x = mu sub y.*0707

*The alternative is that they are different from each other in some way.*0712

*Then we pick a significance level.*0718

*How different do these two populations have to be for us to say they are different?*0721

*We set a decision stage, but instead of drawing the SDOM now we draw the SDOD.*0726

*Because now we are looking at the differences between these to means.*0734

*We identify critical limits and rejection regions.*0739

*We also find the critical test statistic, the boundaries.*0743

*In order to do this we have to find the degrees of freedom for the difference.*0747

*We cannot just use the degrees of freedom for 1, degrees of freedom for the other but we actually add them together.*0753

*And then use the samples and the SDOD to compute the mean difference.*0759

*We are not just computing mean, but we are computing mean difference test statistics, as well as the p value.*0764

*And then we compare the sample to the hypothesized population.*0773

*We either reject the null or not.*0779

*We reject the null if our test statistic and p value lie in those zones of rejection.*0781

*It is like these are the weirdo zone.*0792

*This is all we know that our sample is really different from this population.*0794

*Let us talk about the different formulas that go along with these steps.*0799

*Remember the first step is going to be, what is the hypothesis, the null hypotheses, as well as the alternative.*0806

*This is not really a formula, but it is helpful to remember that this is what we really mean versus x bar – y bar does not equal 0.*0817

*This is often what is going to be the case and you can rewrite this as mu sub x bar – mu sub y bar sometimes,*0836

*but there are some mathematical ideas that you have to learn before you can write that.*0846

*I will leave that aside for now.*0857

*Second thing is significance level.*0859

*Here there are no formulas but you should know that when we say alpha= .05 we are talking about that false alarm rate.*0862

*This is the rate of rejecting the null when the null is actually true.*0873

*This is a very low rate of false alarms.*0877

*When we say alpha = .05 it is not that we calculated it but it is just that*0881

*by convention science tends to say this is the reasonable level of significance.*0887

*Sometimes people are more conservative than 1.0 or 1.001.*0895

*Number 3, we need to set that decision stage.*0900

*It is helpful to draw the SDOD and it is helpful to have our hypothesized population here.*0905

*Mu sub x bay – y bar = 0.*0924

*We assume that this point is 0.*0930

*One thing you probably also want to know about the SDOD is the formula for standard error.*0932

*The formula for standard error of the SDOD we written this a lot of times,*0941

*is the variance of x / n sub x + the variance of y / n sub y.*0951

*Another thing, you probably want to know is that we need to find these critical t.*0959

*We need to find the t values here and in order to find that you will need to know*0965

*the degrees of freedom for the difference and it is pretty easy.*0973

*It is the degrees of freedom for x + the degrees of freedom for y.*0979

*To find this, it is n sub x -1.*0983

*To find that it is n sub y -1.*0988

*We could write this as n sub x -1 + n sub y -1.*0990

*You could write it like that and then I think that is all you need to know for the decision stage.*1002

*Step 4, if you have to compute the samples mean difference you need to calculate its test statistic as well as its p value.*1011

*Remember we are going to be using t from here on out because obviously we are using s instead of sigma.*1039

*Let us talk about how to come to the sample t.*1046

*Let me write this as sample t.*1050

*The sample t is really the distance between where our sample differences versus the hypothesized difference.*1058

*We do not want it just in terms of that raw distance, we want in terms of the standard error.*1069

*It is going to be whatever our x bar - y bar is the actual sample difference -0.*1075

*That is our hypothesized population divided by the standard error s sub x bar – y bar.*1085

*That will give you how many standard errors away our actual mean difference is from 0.*1097

*Once you have this t value and you have the degrees of freedom,*1104

*then you can find the p value and then you could reject or accept the null hypotheses.*1113

*Reject or do not reject, that is really the technical idea there.*1121

*Let us go onto some examples.*1126

*The Cheesy Cheesy cookies company wanted to know whether they should have a coarse or fine texture in their cheesy cookies.*1131

*They assembled a series of taste testing panels that tasted either the coarse*1140

*or fine textured cookies and gave it a palatability score.*1143

*The higher score the better.*1153

*Is there a statistical difference in the mean palatability score between the two texture levels?*1154

*If you download the examples below and you look under the example 1, you should see a data set that looks like this.*1162

*This is the palatability score and this is the texture.*1174

*I believe that 0 = coarse and 1= fine, just so that we can make some sort of recommendation at the end.*1177

*Here we go, we have these different sets of scores, so this is the score that*1200

*one panel came up with and that panel tasted coarse textured cheesy cookies.*1209

*This panel also tasted coarse and that is the score it gave it.*1214

*Let us go up to fine.*1221

*They tasted fine texture and they give it that score.*1223

*They also tasted fine and they give it that score.*1227

*You could go and see what the different scores are and what texture they had.*1231

*First, let us think about what our x and y?*1240

*What are our two independent samples?*1245

*The two independent samples here seem to come from the two different textures.*1247

*One group of scores they all tasted coarse texture cheesy cookies.*1251

*The other group of scores tasted fine textured cheesy cookies.*1260

*It might be helpful to us to sort this data by texture.*1264

*I am going to take this and I am going to ask.*1270

*It would work if I move score over.*1281

*What I am going to do is just hit sort.*1291

*Here these are all our coarse cheesy cookie, the palatability scores and here are my fine cheesy cookie palatability scores.*1296

*Let us think about how we want to approach this problem.*1311

*First thing we want to do is create some sort of hypothesize population.*1315

*Our hypothesize population is really going to say that the coarse and*1322

*fine textured cheesy cookies there is really no difference between them.*1327

*They are the same.*1330

*The mu sub x bar - y bar should equal 0.*1332

*The alternative is that they are different from each other in some way.*1337

*We do not know which one taste better.*1346

*Let us just be neutral and say we do not know whether the coarse cheesy cookies*1352

*are better than the fine or to fine cheesy cookies are better than the coarse.*1358

*We want to know whether these palatability scores are different or the same.*1364

*Let us set a significance level for how different they have to be.*1370

*Our significance level could be alpha= .05.*1377

*Finally let us set a decision stage.*1386

*Here I am going to draw SDOD, can we assume normality?*1390

*Well, they are different and let us look here.*1398

*We have 8 scores and 8 scores, the n is low.*1405

*Technically, we might not be able to do hypothesis testing.*1416

*Let us say for some reason that your teacher wants you doing anyway.*1424

*But one of the things that should come up when you see low n like this is that you should question*1430

*whether hypothesis testing is the right way to go because it may not reflect the conditions*1436

*that we need to have set before we can assume all the stuff.*1446

*Just for the problem solving and practice here, let us go with that.*1449

*But if you want it to be smaller you can tell your instructor the conditions are meet for hypothesis testing.*1454

*Here we set our little lower n rejection and why do we just go ahead and put in our mu here.*1466

*It is going to be 0 and it will be helpful to find out that t values out here.*1478

*Let us go ahead and do that.*1483

*What are our critical t?*1486

*Critical t or the boundaries.*1491

*In order to find the critical t, we are going to have to find the degrees of freedom, DF of differences.*1494

*N sub x we will call x coarses.*1503

*X will be coarse cheesy cookies and y will be fine.*1512

*You can use c and f if you want to.*1521

*This is going to be 8 and this is also 8.*1524

*The degrees of freedom for each of these is 7 so this is going to be 14.*1528

*That is a pretty low degrees of freedom.*1534

*That is all we can assume normality here.*1537

*Let us find the critical t.*1540

*In order to find that we would use t inverse because we have the two tailed probability .05 and we have the degrees of freedom.*1545

*This gives us a positive version.*1562

*The negative version would just be the negative of that number because they are perfectly symmetrical.*1565

*2.14 the critical t is + or -2.14.*1573

*Now that we have that, then we could go ahead and look at the actual samples themselves.*1581

*Step 4, is we need to find the samples mean difference.*1589

*We need to find x bar – y bar, but we also need to find this mean differences t.*1598

*The t sub x bar - y bar.*1606

*We need to find that as well as the p value.*1610

*Let us go ahead and do that.*1613

*We just started from step 3 and step 4 is really the mean difference and that is just the average of these guys - the average of these guys.*1618

*That is their average difference.*1656

*This is saying that the coarse scores tend to be on average lower than*1662

*the fine scores because we do course score – fine score.*1668

*We get a negative number.*1671

*The coarse score number must have been small.*1672

*Actually before we go on, it might be helpful to find the standard error of this situation.*1677

*In order to find the standard error of the difference we need to find*1690

*the square roots of the variance of x ÷ n sub x + the variance of y ÷ n sub y.*1699

*This is going to be our standard error that we need.*1717

*In order to find that it would be helpful to find each of these pieces by themselves.*1724

*I guess we could find the whole thing, the variance of x ÷ n sub x and the variance of y ÷ n sub y.*1731

*I will put each of these on different lines like we can do all of it together.*1750

*We could just add them all up here.*1754

*Let us find that.*1757

*The variance, thankfully Excel has all these functions.*1763

*Let us check and make sure that this variance will give us n-1.*1771

*The variance of x ÷ 8 and the variance of all my fine cheesy cookie values ÷ 8.*1778

*We have these two variances and when we divide by n sub x we are getting the variance of the SDOM.*1799

*If we add those together then get the square root, then we get the standard error of the difference.*1811

*The square root of these two guys added together and that is 11.16.*1820

*Here I will just add this information so the standard error of the difference =11.16.*1830

*In order to find this t, we need to have this difference between the means -0 / the standard error of the difference.*1851

*We can easily do that now.*1866

*Here in order to find the sample t we could put the mean difference -0.*1871

*If you want to keep it technical you do not need that -0 / the standard error of the difference.*1891

*Our sample t says the difference is not at 0 it is actually way down here.*1901

*It is not significantly different.*1914

*Well, one thing we could do is just operate here and compare this number to this number.*1917

*This sub boundary here is -2.14.*1923

*-4.73 is like out here so we definitely know it is way significant.*1928

*It is way standing out from the expected mean but we can also find the p value.*1935

*Now remember in Excel one of the things it needs a positive t value.*1944

*If you have a negative t value you have to turn it into a positive one, but it is okay because it is perfectly symmetrical.*1951

*The degrees of freedom that we are talking about are going to be this*1959

*new combined degrees of freedom because we are always talking that the SDOM now.*1963

*This is the degrees of freedom for this SDOD and that is 14 and it is a two-tailed hypothesis.*1969

*Our p value is .0003.*1976

*I will not write the last up here but we can just talk about it.*1981

*The last step would be we reject or do not reject the null.*1991

*Well, we reject the null here because our t value is much lower than our significance level.*1997

*Our t value, our sample t is more extreme than our critical t.*2003

*Here what we would say is that there is a statistical difference between the two texture levels.*2010

*One that is very unlikely to be attributed to by chance, because that is what this t values.*2018

*If it was by chance it would have .03% probability.*2026

*It is pretty low.*2033

*Example 2, scientists have found certain tree resins that are deadly to termites.*2035

*To test the protective power of resin protecting the tree, a lab prepared 16 dishes with 25 termites in each.*2042

*Each dish was randomly assigned to be treated with 5 mg or 10 mg of resin.*2050

*At the end of 15 days, the number of surviving termites was counted.*2055

*Assume that termites survival tends to be normally distributed with both dosage levels.*2060

*Is there a statistical significant difference in the mean number of survival for those two doses?*2066

*Now here I think it is worth than just discussing what will be our x and y.*2072

*Our x might be the 5 mg population and our y might be the 10 mg population.*2077

*The n sub x some people might think there are 25 termites but actually there is 25 termites in each of 10 Peachtree dishes.*2087

*There are 8 Peachtree dishes that have been randomly treated with 5 mg and 8 have been treated with 10 mg.*2099

*This is 8 and 8.*2109

*When I say 8, we mean the dishes of treatment and the termites are not the subject they are the cases that we are interested in.*2113

*The termites are the test.*2124

*You can get 25 termites surviving or you could get 0 surviving.*2128

*How many termites survived?*2134

*That is our dependent variable.*2135

*Okay, let us see.*2137

*Well one thing we could do is start off with our hypotheses.*2142

*Our null hypotheses is that these two dosage levels are roughly the same.*2146

*We might say something like the mu sub x bar - y bar which is equal 0 are the same.*2153

*The alternative is that they are not the same.*2161

*Maybe that one is more powerful than the other.*2166

*We do not know which one.*2169

*We could easily set our significance level to be .05.*2173

*Let us talk about the actual set up, the decision stage.*2179

*In the decision stage, let us see what we have here.*2184

*We have set up this .05 level rejection and we could just go ahead and this is the x bar - y bar, but what would be that t?*2195

*The nice thing about this being 0 is that the t distribution as well as the x bar – y bar start off the same.*2213

*They are not going to have the same numbers out here.*2226

*Okay, so that is why we do have to put them on different lines.*2229

*They are still talking about different things.*2233

*Let us talk about the t values.*2235

*Before we do, it might be helpful to figure out the new degrees of freedom.*2240

*The degrees of freedom of differences will be 7 + 7 =14.*2247

*Here we can do hypothesis testing just jump in right away because given*2255

*the termite survival tends to be normally distributed within these two dosage rates.*2261

*If you go to example 2, you will actually see the data here.*2267

*Here we see dosage and here is the 5 mg, as well as the 10 mg.*2284

*Here are the survival counts.*2293

*How many termites survived?*2294

*Notice that there is no survival count over 25.*2296

*25 is the maximum you can have, but even the highest gives me 16.*2299

*What if the survival count cannot go below 0 because we cannot have negative termite surviving.*2304

*Here we have the survival count.*2311

*Let us see what we have here.*2317

*Can we figure out what the critical t is.*2323

*Can we figure out what the critical t is?*2329

*I think we can.*2335

*Let us see.*2336

*You can use the book but I am going to use Excel to find the critical t.*2338

*I am going to write for myself step 4.*2344

*I know the two-tailed probability that I need .05 and I know my degrees of freedom is 14.*2347

*I see that the critical t is the same as before and because we use*2362

*the same two tailed probability and the same degrees of freedom of differences.*2367

*Here we know that it is -2.14, as well as positive 2.14.*2372

*What we can do is now from here go on to looking at our actual sample.*2384

*This is actually step 3, it is a part of our decision stage.*2394

*Step 4, is now actually talking about the sample.*2406

*It will help to find the sample mean difference, so that is going to be the average of one of these x - the average y.*2410

*We want to know is this is difference going to be significantly different from 0?*2431

*We cannot just look at the raw scores because we need to figure out how many standard errors away we are.*2436

*How shall we find the standard error for the difference?*2443

*That is equal to the square root of the variance of x/ n sub x + variance of y/ n sub y.*2448

*Let us find the variance of x/ n sub x over and variance of y/ n sub y.*2458

*Let us find the variance of x/8 and the variance of y /8.*2468

*We see that the variance for y is a lot different than the variance for x.*2486

*That is helpful for us to just look at briefly right now just because this will probably give us an idea*2493

*that the variance of samples are so different we probably do not have a good reason to pull these two together.*2500

*We do not have a good reason to assume that the populations are similar.*2507

*When in doubt go with non homogenous variances.*2511

*Just assume that they are different.*2518

*Once we have that then we can find the square root of adding these two standard errors together and we get 2.5.*2520

*Once we have all of that then we can find the samples mean difference t.*2535

*And that would be the samples mean difference -0 divided by the standard error of the SDOD.*2548

*What would that be?*2572

*That would be this guy and I am going to leave that subtract 0 part divided by the standard error and we get to 2.15.*2575

*We are close but it is still more extreme than 2.14.*2586

*It does not have to be extreme and the -n could be either extreme in the negative n or extreme in the positive n.*2595

*This is extreme in the positive n.*2603

*It is just right outside our borders.*2607

*Let us find the p value.*2609

*In order to find that p value we use t distribution because we have the t value that*2611

*we want the degrees of freedom and we wanted to be a two-tailed p value.*2620

*It is going to add up this little chunk and this little chunk together and that can be .049.*2625

*We will just skip step 4, our p value =.0449 that is right just a hair underneath our alpha.05.*2635

*We would probably reject the null.*2653

*Example 3, 2 months before smoking ban in bars, a random sample of bar employees were assessed on respiratory health.*2657

*Two months after the ban, another random sample of employees were assessed.*2672

*Researchers saw a statistically significant increase in the mean scores of health.*2678

*P= .049 we had an example of that two tailed.*2684

*Which of the following is the best interpretation for this result?*2689

*The probability is only .049 that the mean score for all of our employees increased from before to after the ban.*2693

*Is that what this means?*2706

*For me it helps to draw that SDOD and it is saying the null hypotheses would be*2708

*the same like before and after are the same.*2715

*What they actually found is that there is some extreme value.*2720

*There is the increase in mean scores.*2727

*There is a positive difference from after – before.*2735

*There is the increased.*2742

*It is somewhere up here, that increase tells us that.*2745

*P= .04.*2749

*We can actually draw this carefully, it is just right above that cut off.*2753

*There is only .049 probability that the mean score for all bar employees increase.*2760

*That is not what this means.*2775

*It is not saying that there is only a small chance that it increase.*2778

*It is actually saying there is a pretty good chance that it is not the same.*2783

*There is a pretty small chance that it is the same.*2787

*This one we can just rule out.*2792

*Another possibility is that the mean score for all bar employees increased by more than 4.9%.*2796

*Does this p value actually talk about the raw score on respiratory health?*2805

*It does not talk about that score at all, it is the probability of finding such a difference.*2814

*It does not have anything to do with actual scores.*2821

*What about this one?*2825

*An observed difference in the sample means as large or larger than the sample is unlikely to occur*2828

*if the mean score for all bar employees before and after the ban were the same.*2835

*This actually have something we can use.*2839

*This is about considering that the means score for before and after are the same.*2842

*That is important because that is what the SDOM actually represents.*2851

*That is what this p value is actually talking something about this idea that when we get the sample,*2854

*we consider that they were just the same.*2865

*This is saying an observed difference in sample means as large or larger than a sample is very unlikely to occur.*2867

*It is likely to occur with .049% if the mean score for all bar employees the true score is actually the same.*2876

*This is a pretty good contender because the SDOD is talking about how .049 means very unlikely.*2889

*This I would leave as a definite contender.*2900

*Maybe there is a better answer.*2902

*There is a 4.9% chance that the mean score of all bar employees after the ban is actually lower than before the ban.*2905

*There is a small chance of the opposite hypotheses picture that is probably not the case.*2915

*It depends on what the null hypothesis was.*2925

*The null hypothesis and a two mean hypotheses test is usually the same not the one is less than the other.*2934

*We do not usually do that.*2953

*Maybe there is a way and that could be true.*2954

*It is probably not true if we did hypothesis testing at all.*2958

*Only 4.9% of the bar employees had their score drop but the other 95% had their scores increase.*2961

*This would be a correct interpretation if we are not talking about the SDOD.*2971

*If this was not a reflection of the population then maybe that would be true.*2977

*This is not talking about population, it is talking about the SDOD.*2982

*This is a wrong interpretation.*2987

*The correct answer is c.*2990

*That is our last example for hypotheses testing with two independent means.*2992

*Thank you for joining us on www.educator.com.*2998

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