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In the next two sections, we explore the use of the t-distribution in more detail—first for computing a confidence interval for a difference in two means and then for computing a p-value to test a hypothesis that two means are equal. SECTION LEARNING GOALS You should now have the understanding and skills to: Use a formula to find the standard error for a distribution of differences in sample means for two groups Recognize when a t-distribution is an appropriate model for a distribution of the standardized difference in two sample means Exercises for Section 6.4-D SKILL BUILDER 1 In Exercises 6.180 to 6.183, random samples of the given sizes are drawn from populations with the given means and standard deviations. For each scenario, use the formula to find the standard error of the distribution of differences in sample means, x?1 - x?2. 6.180 Samples of size 100 from Population 1 with mean 87 and standard deviation 12 and samples of size 80 from Population 2 with mean 81 and standard deviation 15 6.181 Samples of size 25 from Population 1 with mean 6.2 and standard deviation 3.7 and samples of size 40 from Population 2 with mean 8.1 and standard deviation 7.6 6.182 Samples of size 50 from Population 1 with mean 3.2 and standard deviation 1.7 and samples of size 50 from Population 2 with mean 2.8 and standard deviation 1.3 6.183 Samples of size 300 from Population 1 with mean 75 and standard deviation 18 and samples of size 500 from Population 2 with mean 83 and standard deviation 22 SKILL BUILDER 2 Use a t-distribution to answer the questions in Exercises 6.184 to 6.187. Assume the samples are random samples from distributions that are reasonably normally distributed, and that a t-statistic will be used for inference about the difference in sample means. State the degrees of freedom used. 6.184 Find the endpoints of the t-distribution with 2.5% beyond them in each tail if the samples have sizes n1 = 15 and n2 = 25. 6.185 Find the endpoints of the t-distribution with 5% beyond them in each tail if the samples have sizes n1 = 8 and n2 = 10. 6.186 Find the proportion in a t-distribution less than -1.4 if the samples have sizes n1 = 30 and n2 = 40. 6.187 Find the proportion in a t-distribution above 2.1 if the samples have sizes n1 = 12 and n2 = 12. 6.4-CI CONFIDENCE INTERVAL FOR A DIFFERENCE IN MEANS In Section 5.2 we see that when a distribution of a statistic is normally distributed, a confidence interval can be formed using Sample Statistic ± z* · SE where z* is an appropriate percentile from a standard normal distribution and SE is the standard error of the statistic.

          In the next two sections, we explore the use of the t-distribution in more detail—first for computing a confidence interval for a difference in two means and then for computing a p-value to test a hypothesis that two means are equal. SECTION LEARNING GOALS You should now have the understanding and skills to: Use a formula to find the standard error for a distribution of differences in sample means for two groups Recognize when a t-distribution is an appropriate model for a distribution of the standardized difference in two sample means Exercises for Section 6.4-D SKILL BUILDER 1 In Exercises 6.180 to 6.183, random samples of the given sizes are drawn from populations with the given means and standard deviations. For each scenario, use the formula to find the standard error of the distribution of differences in sample means, x?1 - x?2. 6.180 Samples of size 100 from Population 1 with mean 87 and standard deviation 12 and samples of size 80 from Population 2 with mean 81 and standard deviation 15 6.181 Samples of size 25 from Population 1 with mean 6.2 and standard deviation 3.7 and samples of size 40 from Population 2 with mean 8.1 and standard deviation 7.6 6.182 Samples of size 50 from Population 1 with mean 3.2 and standard deviation 1.7 and samples of size 50 from Population 2 with mean 2.8 and standard deviation 1.3 6.183 Samples of size 300 from Population 1 with mean 75 and standard deviation 18 and samples of size 500 from Population 2 with mean 83 and standard deviation 22 SKILL BUILDER 2 Use a t-distribution to answer the questions in Exercises 6.184 to 6.187. Assume the samples are random samples from distributions that are reasonably normally distributed, and that a t-statistic will be used for inference about the difference in sample means. State the degrees of freedom used. 6.184 Find the endpoints of the t-distribution with 2.5% beyond them in each tail if the samples have sizes n1 = 15 and n2 = 25. 6.185 Find the endpoints of the t-distribution with 5% beyond them in each tail if the samples have sizes n1 = 8 and n2 = 10. 6.186 Find the proportion in a t-distribution less than -1.4 if the samples have sizes n1 = 30 and n2 = 40. 6.187 Find the proportion in a t-distribution above 2.1 if the samples have sizes n1 = 12 and n2 = 12. 6.4-CI CONFIDENCE INTERVAL FOR A DIFFERENCE IN MEANS In Section 5.2 we see that when a distribution of a statistic is normally distributed, a confidence interval can be formed using Sample Statistic ± z* · SE where z* is an appropriate percentile from a standard normal distribution and SE is the standard error of the statistic.

In the next two sections, we explore the use of the t-distribution in more detail—first for computing a confidence interval for a difference in two means and then for computing a p-value to test a hypothesis that two means are equal. SECTION LEARNING GOALS You should now have the understanding and skills to: Use a formula to find the standard error for a distribution of differences in sample means for two groups Recognize when a t-distribution is an appropriate model for a distribution of the standardized difference in two sample means Exercises for Section 6.4-D SKILL BUILDER 1 In Exercises 6.180 to 6.183, random samples of the given sizes are drawn from populations with the given means and standard deviations. For each scenario, use the formula to find the standard error of the distribution of differences in sample means, x?1 - x?2. 6.180 Samples of size 100 from Population 1 with mean 87 and standard deviation 12 and samples of size 80 from Population 2 with mean 81 and standard deviation 15 6.181 Samples of size 25 from Population 1 with mean 6.2 and standard deviation 3.7 and samples of size 40 from Population 2 with mean 8.1 and standard deviation 7.6 6.182 Samples of size 50 from Population 1 with mean 3.2 and standard deviation 1.7 and samples of size 50 from Population 2 with mean 2.8 and standard deviation 1.3 6.183 Samples of size 300 from Population 1 with mean 75 and standard deviation 18 and samples of size 500 from Population 2 with mean 83 and standard deviation 22 SKILL BUILDER 2 Use a t-distribution to answer the questions in Exercises 6.184 to 6.187. Assume the samples are random samples from distributions that are reasonably normally distributed, and that a t-statistic will be used for inference about the difference in sample means. State the degrees of freedom used. 6.184 Find the endpoints of the t-distribution with 2.5% beyond them in each tail if the samples have sizes n1 = 15 and n2 = 25. 6.185 Find the endpoints of the t-distribution with 5% beyond them in each tail if the samples have sizes n1 = 8 and n2 = 10. 6.186 Find the proportion in a t-distribution less than -1.4 if the samples have sizes n1 = 30 and n2 = 40. 6.187 Find the proportion in a t-distribution above 2.1 if the samples have sizes n1 = 12 and n2 = 12. 6.4-CI CONFIDENCE INTERVAL FOR A DIFFERENCE IN MEANS In Section 5.2 we see that when a distribution of a statistic is normally distributed, a confidence interval can be formed using Sample Statistic ± z* · SE where z* is an appropriate percentile from a standard normal distribution and SE is the standard error of the statistic.

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