Inference About Means and Proportions with Two Populations. Chapter 10

Transcription

1 Inference About Means and Proportions with Two Populations Chapter 10

2 Two Populations? Chapter 8 we found interval estimates for the population mean and population proportion based on a random sample Chapter 9 we performed hypothesis tests for the population mean and population proportion based on a random sample

3 Two Populations? Want to determine what percentage of middle school students have their own cell phone. Want to determine if the average number of vacation days taken per year has increased. Want to determine the difference in annual income between workers who ve earned a bachelor s degree and workers who haven t earned a bachelor s degree. Want to compare the relapse rate between patients who undergo treatment A, and patients who undergo treatment B.

4 Inference About Means with Two Populations If μ 1 is the mean of population 1 and μ is the mean of population, then the difference between the population means is μ 1 μ. To estimate this difference, independently create a simple random sample of n 1 elements from population 1, and a simple random sample of n elements from population. Compute the means for each sample, x 1 and x. x 1 x is our point estimate for μ 1 μ.

5 Inference About Means with Two Populations Standard Error of x 1 x is σ x 1 x = σ 1 + σ n 1 If both populations are normally distributed, or if the sample sizes are large enough (n 1 30 and n 30), then the differences between the sample means will have a normal distribution, centered about μ 1 μ. n

6 Interval Estimate of the Difference between two Population Means (σ 1 and σ known) So our (1 α) confidence interval for μ 1 μ is: σ x 1 x ± z 1 α/ + σ n 1 n If both populations are normally distributed, or if the sample sizes are large enough (n 1 30 and n 30), then the differences between the sample means will have a normal distribution, centered about μ 1 μ. Note: z α/ σ 1 + σ n 1 n is called the margin of error for our estimate.

7 Example Super Tread tire company believes that their new Treadtastic tire will hold up better than competitor Treadmaster s Treadful tire. To compare tread wear for the tires, a sample of Treadtastic tires was compared with at sample of Treadful tires.

8 Example Based on the data below, we would like to determine a 95% confidence interval for how much further a Treadtastic tire will last compared to a Treadful tire before needing to be replaced. Treadtastic Treadful Sample Size 100 tires 85 tires Sample Mean 5,386 miles 49,748 miles Know population standard deviation 1,100 miles 980 miles

9 Example n 1 = 100, n = 85, x 1 = 5386, σ 1 = 1100 x = 49748, σ = 980 Treadtastic Treadful Sample Size 100 tires 85 tires Sample Mean 5,386 miles 49,748 miles Know population standard deviation 1,100 miles 980 miles

10 Interval Estimate of the Difference between two Population Means (σ 1 and σ known) n 1 = 100, n = 85, x 1 = 5386, σ 1 = 1100 x = 49748, σ = 980 z α/ =1.96 x 1 σ x ± z 1 α/ + σ n 1 n

11 Example Our % confident that the difference between the mean distance travelled before replacement for Treadtastic tires and Treadful tires is between miles and miles.

12 Hypothesis Tests About μ 1 μ (σ 1 and σ known) H 0 : μ 1 μ D 0 OR H 0 : μ 1 μ D 0 OR H 0 : μ 1 μ = D 0 H a : μ 1 μ < D 0 H a : μ 1 μ > D 0 H a : μ 1 μ D 0 The test statistic: z = ( x 1 x ) D 0 σ 1 n1 +σ n

13 Hypothesis Test for μ 1 μ (σ 1 and σ known) State the hypotheses: H 0 : μ 1 μ D 0 OR H 0 : μ 1 μ D 0 OR H 0 : μ 1 μ = D 0 H a : μ 1 μ < D 0 H a : μ 1 μ > D 0 H a : μ 1 μ D 0 Draw a picture Compute the test statistic:z = ( x 1 x ) D 0 σ 1 n1 +σ n Compute/Estimate the p-value Lower: area to left of t Upper: area to right of t Two-Tailed: t > 0, twice area to right t < 0, twice area to left Decide whether or not to reject H 0. Reject if p-value α

14 Example: Hypothesis Tests About μ 1 μ (σ 1 and σ known) Can we say with a 1% level of significance that the mean distance that Treadtastic tires travel before needing replacement is greater than the mean distance that Treadful tires travel before needing replacement? H 0 : μ 1 μ 0 H a : μ 1 μ > 0

15 Example: Hypothesis Tests About μ 1 μ (σ 1 and σ known) z = ( x 1 σ 1 x ) D 0 + σ n 1 n p value = Is the p-value less than α? Treadtastic. = Treadful Sample Size 100 tires 85 tires Sample Mean 5,386 miles 49,748 miles Know population standard deviation 1,100 miles 980 miles

16 Problem 7 on page 414 of your textbook (Anderson, Sweeny, Williams. Statistics for Business and Economics, 11e) During the 003 season, Major League Baseball took steps to speed up the play of games in order to maintain fan interest (CNN Headline News, September 30, 003). For a sample of 60 games played during the summer of 00, the mean game duration was hours and 5 minutes. For a sample of 50 games played during the summer of 003, the mean game duration was hours and 46 minutes. a. A research hypothesis was that the steps taken would reduce the mean game duration. State the null and alternative hypotheses. b. Compute the point estimate for reduction in game duration. c. Assume a population standard deviation of 1 minutes for both years, determine the p-value. Using a 0.05 level of significance, what is your conclusion? d. Provide a 95% confidence interval estimate on the reduction in mean game duration. e. Should management be pleased?

17 Interval Estimate of the Difference between two Population Means (σ 1 and σ unknown) The (1 α) confidence interval for μ 1 μ is: s x 1 x ± t 1 α/ + s Margin of Error n 1 n The degrees of freedom for a t distribution with two independent random samples is given by df = 1 n1 1 s 1 n1 +s n s n1 n 1 s n (round non-integer values down)

18 Example Employed adults in Charlotte and Rock Hill were sampled to determine the difference in the average amounts two populations spend on lunch each day. 40 employed adults in Charlotte were sampled. The sample mean was $11.5 with a sample standard deviation of $ employed adults in Rock Hill were sampled. The sample mean was $9.76 with a sample standard deviation of $3.04. Find a 90% confidence interval for the difference in the means for the two populations.

19 Hypothesis Tests About μ 1 μ (σ 1 and σ unknown) H 0 : μ 1 μ D 0 OR H 0 : μ 1 μ D 0 OR H 0 : μ 1 μ = D 0 H a : μ 1 μ < D 0 H a : μ 1 μ > D 0 H a : μ 1 μ D 0 The test statistic: t = x 1 x D 0 s 1 n1 +s n Note: when possible, use equal sample sizes and have at least 0 elements total (across the two samples).

20 Example Employed adults in Charlotte and Rock Hill were sampled to determine the difference in the average amounts two populations spend on lunch each day. 40 employed adults in Charlotte were sampled. The sample mean was $11.5 with a sample standard deviation of $ employed adults in Rock Hill were sampled. The sample mean was $9.76 with a sample standard deviation of $3.04. Can we state with a 1% level of significance that the average amount spent for lunch by employed Charlotteans is more than the average amount spent by employed Rock Hillians