1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 23 Inference About Means

2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 23- 2 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 The Student’s t-models form a whole of related distributions that depend on a parameter known as We often denote degrees of freedom as, and the model as Student’s t. family degrees of freedom. df t df.

3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 23- 3 What Does This Mean 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 degrees of freedom. We estimate the standard error with n – 1

4. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 23- 4 What Does This Mean for Means? (cont.) Student’s t-models are,, and, just like the Normal. But t-models with only a few degrees of freedom have than the Normal. unimodal symmetric bell shaped much fatter tails

5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 23- 5 What Does This Mean for Means? (cont.) As the degrees of freedom, the t-models look more and more like the Normal. In fact, the t-model with infinite degrees of freedom is exactly Normal. increase

6. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 23- 6 Finding t-Values By Hand The Student’s t-model is 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. different

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 23- 7 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. Use your calculator to find the following. a)P(t > 1.645) df = 12b) P(t < 0.53) df = 23 c)t-values for 90% CId) t-values for 95% CI df = 6 df = 35 0.0629 0.6994 ± 1.943 ± 2.030

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 23- 8 Assumptions and Conditions Independence Assumption: Randomization Condition: The data arise from a or Randomly sampled data (particularly from an SRS) are ideal. 10% Condition: When a sample is drawn, the sample should be no more than 10% of the population. random sample suitably randomized experiment. without replacement

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 23- 9 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 and Check this condition by making a or unimodalsymmetric. histogram Normal probability plot.

10. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 23- 10 Assumptions and Conditions (cont.) Nearly Normal Condition: The smaller the sample size ( ), the more closely the data should follow a Normal model. For moderate sample sizes ( ), the t works well as long as the data are unimodal and reasonably symmetric. For larger sample sizes, the t methods are safe to use even if the data are skewed. n < 15 or so n between 15 and 40

11. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 23- 11 A nutrition laboratory tests 40 “reduced sodium” hot dogs, finding that the mean sodium content is 310 mg, with a standard deviation of 36 mg. a) Find a 95% confidence interval for the mean sodium content of this brand of hot dog. b) What assumptions have you made in this inference? Are the appropriate conditions satisfied? c) Explain clearly what your interval means. (298.49, 321.51) Population is unimodal and reasonably symmetric, random sample. 40 hotdogs is less than 10% of all these types of hotdogs. Also check the nearly normal condition with graph. We are 95% confident that the true mean of sodium content for these types of hotdogs is between 298.49 and 321.51 mg

12. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 23- 12 More Cautions About Interpreting Confidence Intervals Remember that interpretation of your confidence interval is key. What NOT to say: “90% of all the vehicles on Triphammer Road drive at a speed between 29.5 and 32.5 mph.” The confidence interval is about the not the “We are 90% confident that a randomly selected vehicle will have a speed between 29.5 and 32.5 mph.” Again, the confidence interval is about the not the mean individual values. mean individual values.

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 23- 13 More Cautions About Interpreting Confidence Intervals (cont.) What NOT to say: “The mean speed of the vehicles is 31.0 mph 90% of the time.” The true mean does not —it’s the confidence interval that would be different had we gotten a different sample. “90% of all samples will have mean speeds between 29.5 and 32.5 mph.” The interval we calculate does not for every other interval—it is no more (or less) likely to be correct than any other interval. vary set a standard

14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Assignment Pg. 541 #1, 5, 8, 9, 12 Slide 23- 14

15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 23 Inference About Means (2)

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 23- 16 One-Sample t-test for the Mean The 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 We use that model to obtain a Student’s t model with n – 1 df. P-value.

17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Ex. 2In 1960 census results indicated that the age at which American men first married had a mean of 23.3 years. It is widely suspected that young people today are waiting to get married. A random sample of 40 men married at and average age of 24.2 years with a standard deviation of 5.3 years. a) Are the assumptions satisfied? b) Test this claim. c) What does the p-value mean in context? Slide 23- 17 Random sample, Population may not be normal (easier to have a few really older men than younger, however since n = 40, then it should approximate normal. Check graph for outliers or severe skewness. H 0 :µ = 23.3 years H A :µ > 23.3 years p-value ≈ 0.1447; therefore we fail to reject the null, with a p-value this big there is not enough evidence to suggest that people today are waiting significantly longer to get married. If the average age at first marriage is still 23.3 years, there is a 14.5% chance of getting a sample mean of 24.2 years or older from a sample of 40 men simply from sampling variation.

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Ex. 3 Psychology experiments sometimes involve testing the ability of rats to navigate mazes. The mazes are classified according to difficulty, as measured by the mean length of time it takes rats to find the food at the end. One researcher needs a maze that will take rats an average of about one minute to solve. He test one maze on several rats, collecting the data that follow. a)Plot the data. Do you think the conditions for inference are satisfied? Explain. b)Test the hypothesis that the mean completion time for this maze is 60 seconds. What is your conclusion? c)Eliminate the outlier, and test the hypothesis again. What is your conclusion? d)Do you think this maze meets the “one-minute average” requirement? Explain. Slide 23- 18 Time (sec) 38.446.2 62.538.0 62.833.9 50.435.0 52.860.1 55.157.6 55.549.5 40.944.3 93.847.9 69.246.2 56.3 Normal Probability Plot fairly straight with one outlier. Ok for inference although outlier may be a problem. H 0 :µ = 60 seconds H A :µ ≠ 60 seconds p-value ≈ 0.0160; Depending on α we may or may not reject null. p-value ≈ 0.0003; without the outlier we would reject the null and conclude that the time is different than 60 seconds. The maze does not meet the “one minute average” requirement.

19. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 23- 19 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.

20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Ex. 4 Suppose we want a margin of error of 3 minutes and we think the standard deviation is 5 minutes based on a pilot study. a)Using α = 0.05 for a one tail test, what is the sample size we should use initially? b)Using degrees of freedom based on this sample size recalculate, the sample size again. Slide 23- 20

21. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 23- 21 Sample Size (cont.) Sample size calculations are exact. The margin of error you find after collecting the data the one you used to find n. The sample size formula depends on quantities you won’t have until you, 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 to give you a good chance of being able to tell you what you want to know. never won’t match exactly collect the data large enough

22. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 23- 22 What have we learned? Statistical inference for means relies on the same concepts as for proportions—only the and the have changed. The reasoning of inference, the need to verify that the appropriate, and the proper interpretation of and all remain the same regardless of whether we are investigating means or proportions. mechanics model assumptions are met confidence intervals P-values

23. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Assignment Pg. 544 #14, 21, 23, 26, 28, 30, 33, 35 Slide 23- 23