2. Copyright © 2009 Pearson Education, Inc. Chapter 23 Inferences About Means

3. Copyright © 2009 Pearson Education, Inc. Slide 1- 3 Objectives: The student will be able to: Perform a t-test for the population mean, to include: writing appropriate hypotheses, checking the necessary assumptions, drawing an appropriate diagram, computing the P- value, making a decision, and interpreting the results in the context of the problem. Compute and interpret in context a t-based confidence interval for the population mean, checking the necessary assumptions.

4. Copyright © 2009 Pearson Education, Inc. Slide 1- 4 Getting Started Now that we know how to create confidence intervals and test hypotheses about proportions, it’d be nice to be able to do the same for means. Just as we did before, we will base both our confidence interval and our hypothesis test on the sampling distribution model. The Central Limit Theorem told us that the sampling distribution model for means is Normal with mean μ and standard deviation

5. Copyright © 2009 Pearson Education, Inc. Slide 1- 5 All we need is a random sample of quantitative data. And the true population standard deviation, σ. Well, that’s a problem… Getting Started (cont.)

6. Copyright © 2009 Pearson Education, Inc. Slide 1- 6 Getting Started (cont.) Proportions have a link between the proportion value and the standard deviation of the sample proportion. This is not the case with means—knowing the sample mean tells us nothing about We’ll do the best we can: estimate the population parameter σ with the sample statistic s (the standard deviation of the sample). Our resulting standard error is

7. Copyright © 2009 Pearson Education, Inc. Slide 1- 7 Getting Started (cont.) We now have extra variation in our standard error from s, the sample standard deviation. We need to allow for the extra variation so that it does not mess up the margin of error and P-value, especially for a small sample. And, the shape of the sampling model changes—the model is no longer Normal. So, what is the sampling model?

8. Copyright © 2009 Pearson Education, Inc. Slide 1- 8 Gosset’s t William S. Gosset, an employee of the Guinness Brewery in Dublin, Ireland, worked long and hard to find out what the sampling model was. The sampling model that Gosset found has been known as Student’s t. The Student’s t-models form a whole family of related distributions that depend on a parameter known as degrees of freedom. We often denote degrees of freedom as df, and the model as t df.

9. Copyright © 2009 Pearson Education, Inc. Slide 1- 9 A Confidence Interval for Means? A practical sampling distribution model for means When the conditions are met, the standardized sample mean follows a Student’s t-model with n – 1 degrees of freedom. We estimate the standard error with

10. Copyright © 2009 Pearson Education, Inc. Slide 1- 10 When Gosset corrected the model for the extra uncertainty, the margin of error got bigger. Your confidence intervals will be just a bit wider and your P-values just a bit larger than they were with the Normal model. By using the t-model, you’ve compensated for the extra variability in precisely the right way. A Confidence Interval for Means? (cont.)

11. Copyright © 2009 Pearson Education, Inc. Slide 1- 11 When the conditions are met, we are ready to find the confidence interval for the population mean, μ. The confidence interval is where the standard error of the mean is The critical value depends on the particular confidence level, C, that you specify and on the number of degrees of freedom, n – 1, which we get from the sample size. A Confidence Interval for Means? (cont.) One-sample t-interval for the mean

12. Copyright © 2009 Pearson Education, Inc. Slide 1- 12 Student’s t-models are unimodal, symmetric, and bell shaped, just like the Normal. But t-models with only a few degrees of freedom have much fatter tails than the Normal. A Confidence Interval for Means? (cont.)

13. Copyright © 2009 Pearson Education, Inc. Slide 1- 13 As the degrees of freedom increase, the t-models look more and more like the Normal. In fact, the t-model with infinite degrees of freedom is exactly Normal. A Confidence Interval for Means? (cont.)

14. Copyright © 2009 Pearson Education, Inc. Slide 1- 14 Assumptions and Conditions Gosset found the t-model by simulation. Years later, when Sir Ronald A. Fisher showed mathematically that Gosset was right, he needed to make some assumptions to make the proof work. We will use these assumptions when working with Student’s t-model.

15. Copyright © 2009 Pearson Education, Inc. Slide 1- 15 Assumptions and Conditions (cont.) Independence Assumption: Independence Assumption. The data values should be independent. Randomization Condition: The data arise from a random sample or suitably randomized experiment. Randomly sampled data (particularly from an SRS) are ideal. 10% Condition: When a sample is drawn without replacement, the sample should be no more than 10% of the population.

16. Copyright © 2009 Pearson Education, Inc. Slide 1- 16 Assumptions and Conditions (cont.) Normal Population Assumption: We can never be certain that the data are from a population that follows a Normal model, but we can check the Nearly Normal Condition: The data come from a distribution that is unimodal and symmetric. Check this condition by making a histogram or Normal probability plot.

17. Copyright © 2009 Pearson Education, Inc. Slide 1- 17 Assumptions and Conditions (cont.) Nearly Normal Condition: The smaller the sample size (n < 15 or so), the more closely the data should follow a Normal model. For moderate sample sizes (n between 15 and 40 or so), the t works well as long as the data are unimodal and reasonably symmetric. For sample sizes larger than 40 or 50, the t methods are safe to use unless the data are extremely skewed.

18. Copyright © 2009 Pearson Education, Inc. Slide 1- 18 Make a Picture, Make a Picture, Make a Picture Pictures tell us far more about our data set than a list of the data ever could. The only reasonable way to check the Nearly Normal Condition is with graphs of the data. Make a histogram of the data and verify that its distribution is unimodal and symmetric with no outliers. You may also want to make a Normal probability plot to see that it’s reasonably straight.

19. Copyright © 2009 Pearson Education, Inc. Finding a confidence interval for the mean when sigma is known Stat -> Tests -> #7 Zinterval Use this ONLY when sigma is known and the population is normal: A sample of 256 residents of a city indicated a monthly mean electric bill of $90 with a standard deviation of $24. Construct a 95% confidence interval for the mean monthly electric bill of all residents. Stat…Tests…#7 for Z Interval…enter For inpt: choose Stats and fill in the rest of the screen with the inputted stats. You should find the interval is $87.06 - $92.94. We are 95% confident that the mean electric monthly bill for residents is between $87.06 and $92.94. Slide 1- 19

20. Copyright © 2009 Pearson Education, Inc. Finding a Confidence Interval for the mean in general (when sigma is unknown) Stat -> Tests -> #8 Tinterval Use this when the population (true) sigma is unknown. Relies on sample mean and sample standard deviation The survival times in weeks are given for 20 male rats which were exposed to a high level of radiation. 152 152 115 109 137 88 94 77 160 165 20 128 123 136 101 62 152 83 69 125 Determine the 95% confidence interval on the mean survival time for rats. First enter your data in L1 (if you have summary stats skip this) Tinterval: Inpt: Data List: L1 Freq: 1 C-level.95 Result:94.57 weeks to 130.23 weeks. Slide 1- 20

21. Copyright © 2009 Pearson Education, Inc. 23. Suppose the elapsed time of airline itineraries between Washington, D.C. and Boston is normally distributed with an unknown population mean and an unknown population standard deviation. Further suppose that a sample of size 25 (therefore, n=25 and degrees of freedom=24) was taken and the following statistics were gotten from the sample: sample mean (ybar) =135 and sample standard deviation (s) = 40. Construct confidence intervals around the sample mean corresponding to the following confidence levels (express the lower and upper bounds of the intervals as INTEGERS): Stat -> Tests -> Tinterval (here we’ll input stats instead of raw data) Inpt: Stats: x(bar): 135, Sx: 40, n:25, C-level: try.80,.90,.95,.98, &.99 a. 80% (124, 146) b. 90% (121, 149) c. 95% (118, 152) d. 98% (115, 155) e. 99% (113, 157) Hint: Your intervals should get wider and wider and should all be centered around 135. Slide 1- 21

22. Copyright © 2009 Pearson Education, Inc. Slide 1- 22 A Test for the Mean The assumptions and conditions for the one-sample t-test for the mean are the same as for the one-sample t-interval. We test the hypothesis H 0 :  =  0 using the statistic The standard error of the sample mean is When the conditions are met and the null hypothesis is true, this statistic follows a Student’s t-model with n – 1 df. We use that model to obtain a P-value. One-sample t-test for the mean

23. Copyright © 2009 Pearson Education, Inc. Slide 1- 23 Finding t-Values By Hand The Student’s t-model is different for each value of degrees of freedom. Because of this, Statistics books usually have one table of t-model critical values for selected confidence levels.

24. Copyright © 2009 Pearson Education, Inc. Slide 1- 24 Finding t-Values By Hand (cont.) Alternatively, we could use technology to find t critical values for any number of degrees of freedom and any confidence level you need. What technology could we use? The Appendix of ActivStats on the CD Any graphing calculator or statistics program

25. Copyright © 2009 Pearson Education, Inc. One sample T-test for the mean - TI-83+ Judy is a ad designer who designs the newspaper ads for the Giant grocery store. Electronic counters at the entrance total the number of people entering the store. Before Judy was hired, the mean number of people entering every day was 3018. Since she has started working at the Giant the management thinks that this average has increased. A random sample of 42 business days gave an average of 3333 people entering the store daily with a standard deviation of 287. Does this indicate that the average number of people entering the store every day has increased? Use an alpha of 0.01. Stat…. Tests…T-Test Inpt: choose data or stats according to what is available (usually you use stats) μ 0 : stands for the hypothecated mean which is in your hypothesis In this case we are testing against the previous 3018. X bar is the sample mean of 3333 Sx is the standard deviation of 287 n: the number in your sample 42 days μ : choose the notation used in the alternative hypothesis In this case we are looking for an increase so choose > Calculate We get a p value of.000000006 which is very, very slight. Therefore, we reject the null hypothesis and conclude that the average number of people entering the store has certainly increased. Slide 1- 25

26. Copyright © 2009 Pearson Education, Inc. 24. Suppose the elapsed time of airline itineraries between Washington, D.C. and Boston is normally distributed with an unknown population mean and unknown population standard deviation. Suppose we randomly sample 25 itineraries and the sample average is calculated to be 135 minutes and the sample standard deviation is calculated to be 40. Further suppose that we want to test the hypothesis that the true population mean (μ) equals 150 minutes. Conduct 2-sided hypothesis tests with the following alpha levels: a. 0.20 b. 0.10c. 0.05 d. 0.02 e. 0.01 H0: µ=150 HA: µ≠150 Test statistic (t) = -1.875 P-value = 0.0730 Conclusions: a. 0.20Reject b. 0.10 Reject c. 0.05 Don’t Reject d. 0.02 Don’t Reject e. 0.01 Don’t Reject Slide 1- 26

27. Copyright © 2009 Pearson Education, Inc. Slide 1- 27 Significance and Importance Remember that “statistically significant” does not mean “actually important” or “meaningful.” Because of this, it’s always a good idea when we test a hypothesis to check the confidence interval and think about likely values for the mean.

28. Copyright © 2009 Pearson Education, Inc. Slide 1- 28 Intervals and Tests Confidence intervals and hypothesis tests are built from the same calculations. In fact, they are complementary ways of looking at the same question. The confidence interval contains all the null hypothesis values we can’t reject with these data.

29. Copyright © 2009 Pearson Education, Inc. Slide 1- 29 Intervals and Tests (cont.) More precisely, a level C confidence interval contains all of the possible null hypothesis values that would not be rejected by a two-sided hypothesis test at alpha level 1 – C. So a 95% confidence interval matches a 0.05 level test for these data. Confidence intervals are naturally two-sided, so they match exactly with two-sided hypothesis tests. When the hypothesis is one sided, the corresponding alpha level is (1 – C)/2.

30. Copyright © 2009 Pearson Education, Inc. 25. Hoping to lure more shoppers downtown, a city builds a new public parking garage. The city plans to pay for the structure through parking fees. During a two month period (44 week days) daily fees collected averaged $126 with a standard deviation of $15. If a consultant claimed that the average daily income would be $130, should we reject her claim using alpha=0.10 (perform a 2-sided test)? H0: µ=130 HA: µ≠130 Test statistic (t): -1.77 P-value: 0.08 Conclusion: Reject the null. It is likely that the consultant’s claim is false. Slide 1- 30

31. Copyright © 2009 Pearson Education, Inc. 26. In 1998, the Nabisco Company announced a “1000 Chips Challenge” claiming that every 18 ounce bag of Chips Ahoy contained at least 1000 chocolate chips. Below are the counts of chips in selected bags. 12191214108712001419112113251345 12441258135611321191127012951135 Perform a one-sided test (H A : µ>1000). What does this evidence say about Nabisco’s claim (let alpha=0.05)? H0: µ=1000 HA: µ>1000 Test statistic (t) = 10.1 P-Value = tiny # Conclusion: We can reject the null hypothesis. It appears that Nabisco’s claim is valid. Slide 1- 31

32. Copyright © 2009 Pearson Education, Inc. 27. When consumers apply for credit, their credit is rated using FICO scores. A random sample of credit ratings is obtained, and the FICO scores are summarized with these statistics: n=18, ybar=660.3, s=95.9. Use an alpha of 0.05 and do a 2-sided hypothesis test to test the claim that the mean credit score (of the general population) is equal to 700 (Triola 2008). H0: µ=700 HA: µ≠700 Test statistic (t) = -1.76 P-Value = 0.097 Conclusion: We cannot reject the null hypothesis. There is not enough statistical evidence to conclude that the mean is different than 700. Slide 1- 32

33. Copyright © 2009 Pearson Education, Inc. 28. Different cereals are randomly selected, and the sugar content is obtained for each cereal, with the results given below for Cheerios, Harmony, Smart Start, Cocoa Puffs, Lucky Charms, Corn Flakes, Fruit Loops, Wheaties, Cap’n Crunch, Frosted Flakes, Apple Jacks, Bran Flakes, Special K, Rice Krispies, Corn Pops, and Trix. Use an alpha of 0.05 to test the claim of a cereal lobbyist that the mean of all cereals is LESS than 0.3 g (Triola 2008). 0.03 0.240.300.470.430.070.470.13 0.44 0.390.480.170.130.090.450.43 H0: µ=0.3 HA: µ<0.3 Test statistic (t): -0.12 P-value: 0.45 Conclusion: Fail to reject the null. The statistical analysis does not back up the lobbyist’s claim. Slide 1- 33

34. Copyright © 2009 Pearson Education, Inc. For Your Information Slide 1- 34

35. Copyright © 2009 Pearson Education, Inc. Slide 1- 35 Sample Size To find the sample size needed for a particular confidence level with a particular margin of error (ME), solve this equation for n: The problem with using the equation above is that we don’t know most of the values. We can overcome this: We can use s from a small pilot study. We can use z* in place of the necessary t value.

36. Copyright © 2009 Pearson Education, Inc. Slide 1- 36 Sample Size (cont.) Sample size calculations are never exact. The margin of error you find after collecting the data won’t match exactly the one you used to find n. The sample size formula depends on quantities you won’t have until you collect the data, but using it is an important first step. Before you collect data, it’s always a good idea to know whether the sample size is large enough to give you a good chance of being able to tell you what you want to know.

37. Copyright © 2009 Pearson Education, Inc. Slide 1- 37 Degrees of Freedom If only we knew the true population mean, µ, we would find the sample standard deviation as But, we use instead of µ, though, and that causes a problem. When we use to calculate s, our standard deviation estimate would be too small. The amazing mathematical fact is that we can compensate for the smaller sum exactly by dividing by which we call the degrees of freedom.

38. Copyright © 2009 Pearson Education, Inc. Slide 1- 38 What Can Go Wrong? Don’t confuse proportions and means. Ways to Not Be Normal: Beware multimodality. The Nearly Normal Condition clearly fails if a histogram of the data has two or more modes. Beware skewed data. If the data are very skewed, try re-expressing the variable. Set outliers aside—but remember to report on these outliers individually.

39. Copyright © 2009 Pearson Education, Inc. Slide 1- 39 What Can Go Wrong? (cont.) …And of Course: Watch out for bias—we can never overcome the problems of a biased sample. Make sure data are independent. Check for random sampling and the 10% Condition. Make sure that data are from an appropriately randomized sample. Interpret your confidence interval correctly. Many statements that sound tempting are, in fact, misinterpretations of a confidence interval for a mean. A confidence interval is about the mean of the population, not about the means of samples, individuals in samples, or individuals in the population.

40. Copyright © 2009 Pearson Education, Inc. Slide 1- 40 What have we learned? Statistical inference for means relies on the same concepts as for proportions—only the mechanics and the model have changed. The reasoning of inference, the need to verify that the appropriate assumptions are met, and the proper interpretation of confidence intervals and P-values all remain the same regardless of whether we are investigating means or proportions.