1. 10.2 Comparing Two Means Objectives SWBAT: DESCRIBE the shape, center, and spread of the sampling distribution of the difference of two sample means. DETERMINE whether the conditions are met for doing inference about µ 1 − µ 2. CONSTRUCT and INTERPRET a confidence interval to compare two means. PERFORM a significance test to compare two means. DETERMINE when it is appropriate to use two-sample t procedures versus paired t procedures.

2. What is meant by “the sampling distribution of the difference between two means?”

3. The Sampling Distribution of the Difference Between Sample Means Choose an SRS of size n 1 from Population 1 with mean µ 1 and standard deviation σ 1 and an independent SRS of size n 2 from Population 2 with mean µ 2 and standard deviation σ 2.

4. However: This measures how far the difference in sample means will typically be from the difference in population means if we repeat the random sampling or random assignment many times.

5. What is the formula for the two-sample t statistic? Is this on the formula sheet? What does it measure? It measures how far the difference in the sample means is from 0, in standardized units.

6. Conditions for Performing Inference About µ 1 - µ 2 Random: The data come from two independent random samples or from two groups in a randomized experiment. o 10%: When sampling without replacement, check that n 1 ≤ (1/10)N 1 and n 2 ≤ (1/10)N 2. Normal/Large Sample: Both population distributions (or the true distributions of responses to the two treatments) are Normal or both sample sizes are large (n 1 ≥ 30 and n 2 ≥ 30). If either population (treatment) distribution has unknown shape and the corresponding sample size is less than 30, use a graph of the sample data to assess the Normality of the population (treatment) distribution. Do not use two-sample t procedures if the graph shows strong skewness or outliers.

7. What distribution does the two-sample t statistic have? Why do we use a t statistic rather than a z statistic? How do you calculate the degrees of freedom? The two-sample t statistic has approximately a t distribution. We can use technology to determine degrees of freedom OR we can use a conservative approach, using the smaller of n 1 – 1 and n 2 – 1 for the degrees of freedom. We use t rather than z when we do not know the population standard deviation.

8. Alternate Example: Leaking Helium After buying many helium balloons only to see them deflate within a couple of days, Erin and Jenna decided to test if helium-filled balloons deflate faster than air-filled balloons. To find out, they bought 60 balloons of the same type and randomly divided them into two piles of 30, filling the balloons in the first pile with helium and the balloons in the second pile with air. Then, they measured the circumference of each balloon immediately after being filled and again three days later. The average decrease in circumference of the helium-filled balloons was 26.5 cm with a standard deviation of 1.92 cm. The average decrease of the air-filled balloons was 2.1 cm with a standard deviation of 2.79 cm. (a)Why was it important that they used the same type of balloons? What is this called in experiments? This is called control. Using the same type of balloon eliminates a possible source of variability, making the standard deviations smaller and increasing the power. b) Do these data provide convincing evidence that helium-filled balloons deflate faster than air- filled balloons?

9. Alternate Example: Leaking Helium After buying many helium balloons only to see them deflate within a couple of days, Erin and Jenna decided to test if helium-filled balloons deflate faster than air-filled balloons. To find out, they bought 60 balloons of the same type and randomly divided them into two piles of 30, filling the balloons in the first pile with helium and the balloons in the second pile with air. Then, they measured the circumference of each balloon immediately after being filled and again three days later. The average decrease in circumference of the helium-filled balloons was 26.5 cm with a standard deviation of 1.92 cm. The average decrease of the air-filled balloons was 2.1 cm with a standard deviation of 2.79 cm.

10. Alternate Example: Leaking Helium After buying many helium balloons only to see them deflate within a couple of days, Erin and Jenna decided to test if helium-filled balloons deflate faster than air-filled balloons. To find out, they bought 60 balloons of the same type and randomly divided them into two piles of 30, filling the balloons in the first pile with helium and the balloons in the second pile with air. Then, they measured the circumference of each balloon immediately after being filled and again three days later. The average decrease in circumference of the helium-filled balloons was 26.5 cm with a standard deviation of 1.92 cm. The average decrease of the air-filled balloons was 2.1 cm with a standard deviation of 2.79 cm. Do: t = 39.46 p = 0 df = 51.4

11. Alternate Example: Leaking Helium After buying many helium balloons only to see them deflate within a couple of days, Erin and Jenna decided to test if helium-filled balloons deflate faster than air-filled balloons. To find out, they bought 60 balloons of the same type and randomly divided them into two piles of 30, filling the balloons in the first pile with helium and the balloons in the second pile with air. Then, they measured the circumference of each balloon immediately after being filled and again three days later. The average decrease in circumference of the helium-filled balloons was 26.5 cm with a standard deviation of 1.92 cm. The average decrease of the air-filled balloons was 2.1 cm with a standard deviation of 2.79 cm. c) Interpret the p-value you got in part (b) in the context of this study. There is approximately a 0 probability of getting a difference of 24.4 cm or more by chance alone, assuming that the mean decrease in circumference is the same for helium-filled and air-filled balloons.

12. Is it OK to use your calculator for the Do step? Are there any drawbacks? We already know we can use it, no partial credit… Note: Make sure you always say “NO” to pooling when doing inference for means.

13. Same conditions as before. Two-Sample t Interval for a Difference Between Two Means

14. Alternate Example: Chocolate Chips Ashtyn and Olivia wanted to know if generic chocolate chip cookies have as many chocolate chips as name-brand chocolate chip cookies, on average. To investigate, they randomly selected 10 bags of Chips Ahoy cookies and 10 bags of Great Value cookies and randomly selected 1 cookie from each bag. Then, they carefully broke apart each cookie and counted the number of chocolate chips in each. Here are their results: Chips Ahoy: 17, 19, 21, 16, 17, 18, 20, 21, 17, 18 Great Value: 22, 20, 14, 17, 21, 22, 15, 19, 26, 18 (a) Construct and interpret a 99% confidence interval for the difference in the mean number of chocolate chips in Chips Ahoy and Great Value cookies.

15. Alternate Example: Chocolate Chips Ashtyn and Olivia wanted to know if generic chocolate chip cookies have as many chocolate chips as name-brand chocolate chip cookies, on average. To investigate, they randomly selected 10 bags of Chips Ahoy cookies and 10 bags of Great Value cookies and randomly selected 1 cookie from each bag. Then, they carefully broke apart each cookie and counted the number of chocolate chips in each. Here are their results: Chips Ahoy: 17, 19, 21, 16, 17, 18, 20, 21, 17, 18 Great Value: 22, 20, 14, 17, 21, 22, 15, 19, 26, 18 Normal/Large Sample: Because there is no obvious skewness or outliers in the graphs below, it is safe to use t procedures.

16. Alternate Example: Chocolate Chips Ashtyn and Olivia wanted to know if generic chocolate chip cookies have as many chocolate chips as name-brand chocolate chip cookies, on average. To investigate, they randomly selected 10 bags of Chips Ahoy cookies and 10 bags of Great Value cookies and randomly selected 1 cookie from each bag. Then, they carefully broke apart each cookie and counted the number of chocolate chips in each. Here are their results: Chips Ahoy: 17, 19, 21, 16, 17, 18, 20, 21, 17, 18 Great Value: 22, 20, 14, 17, 21, 22, 15, 19, 26, 18 Calculator: 2-SampTInt List1: List 2: Freq1: 1 Freq2: 1 C-Level: 0.99 Pooled: No (-4.814, 2.814) df = 13.145

17. Alternate Example: Chocolate Chips Ashtyn and Olivia wanted to know if generic chocolate chip cookies have as many chocolate chips as name-brand chocolate chip cookies, on average. To investigate, they randomly selected 10 bags of Chips Ahoy cookies and 10 bags of Great Value cookies and randomly selected 1 cookie from each bag. Then, they carefully broke apart each cookie and counted the number of chocolate chips in each. Here are their results: Chips Ahoy: 17, 19, 21, 16, 17, 18, 20, 21, 17, 18 Great Value: 22, 20, 14, 17, 21, 22, 15, 19, 26, 18 Conclude: We are 99% confident that the interval from −4.814 to 2.814 captures the true difference in the mean number of chocolate chips in Chips Ahoy and Great Value cookies. b) Does your interval provide convincing evidence that there is a difference in the mean number of chocolate chips? Because the interval includes 0, there is not convincing evidence that there is a difference in the mean number of chocolate chips in Chips Ahoy and Great Value chocolate chip cookies.

18. When doing two-sample t procedures, should we pool the data to estimate a common standard deviation? Is there any benefit? Are there any risks? Do not pool. Ever. Pooling assumes the two populations have the same variance and that the population distributions are exactly Normal. In the real world, distributions are not exactly Normal, and the population variances are not exactly equal. Therefore, do not pool. Note: The benefits to pooling are that we have more degrees of freedom (n-2), which leads to narrower confidence intervals and smaller p-values. However, this is only when population variances are equal and population distributions are exactly Normal. As seen above, this is rarely the case. DON’T POOL!!! What about a two-sample test for a difference in proportions? Why do we pool for this test? We pool for proportions because we need a way to estimate p. We don’t have to worry about that with means.

19. Should you use two-sample t procedures with paired data? Why not? How can you know which procedure to use? Do not use two-sample t procedures on paired data. Think about it this way: Would the test be messed up if the data were scrambled in the lists? If the answer is yes, then the data are paired and you should be using the differences. When determining your procedure, pay attention to the context of the problem. It is going to indicate what procedure you want to use.

20. Alternate Example: Testing with distractions Suppose you are designing an experiment to determine if students perform better on tests when there are no distractions, such as a teacher talking on the phone. You have access to two classrooms and 30 volunteers who are willing to participate in your experiment. a) Design an experiment so that a two-sample t test would be the appropriate inference method. On 15 index cards write “A” and on 15 index cards write “B.” Shuffle the cards and hand them out at random to the 30 volunteers. All 30 subjects will take the same reading comprehension test. Subjects who receive A cards will go to a classroom with no distractions, and subjects who receive B cards will go to a classroom that will have the proctor talking on the phone during the test. At the end of the experiment, compare the mean score for subjects in room A with the mean score for subjects in room B.

21. Alternate Example: Testing with distractions Suppose you are designing an experiment to determine if students perform better on tests when there are no distractions, such as a teacher talking on the phone. You have access to two classrooms and 30 volunteers who are willing to participate in your experiment. b) Design an experiment so that a paired t test would be the appropriate inference method. Using the same procedure in part (a), divide the subjects into two rooms and give them the same reading comprehension test. One room will be distraction free, and the other room will have a proctor talking on the phone. Then, after a short break, give all 30 subjects a similar reading comprehension test but have the distraction in the opposite room. At the end of the experiment, calculate the difference in the two reading comprehension scores for each subject and compare the mean difference with 0.

22. Alternate Example: Testing with distractions Suppose you are designing an experiment to determine if students perform better on tests when there are no distractions, such as a teacher talking on the phone. You have access to two classrooms and 30 volunteers who are willing to participate in your experiment. c) Which experimental design is better? Explain. The experimental design in part (b) is better since it eliminates an important source of variability – the reading comprehension skills of the individual subjects. MORE POWER! d) What is the purpose of random assignment in this experiment? To make sure one environment (treatment) isn’t favored by always going first (or second).