Exam 12: Hypothesis Testing for Two Populations

If the populations are normally distributed and the variances of the two populations are $\textbf{ not }$ equal, it is appropriate to use the "unpooled" formula to determine the t statistic for the hypothesis test of the difference in the two population means.

Suppose that the proportion of young adults who read at least one book per month is 0.15, and this proportion is the same in Boston and New York. Suppose that samples of size 400 are randomly drawn from each city. The standard deviation for the sampling distribution of differences between the first sample proportion and the second sample proportion (used to calculate the z score) is ___.

Maureen McIlvoy, owner and CEO of a mail order business for wind surfing equipment and supplies, is reviewing the order filling operations at her warehouses. Her goal is 100% of orders shipped within 24 hours. In previous years, neither warehouse has achieved the goal, but the East Coast Warehouse has consistently out-performed the West Coast Warehouse. Her staff randomly selected 200 orders from the West Coast Warehouse (population 1) and 400 orders from the East Coast Warehouse (population 2), and reports that 190 of the West Coast Orders were shipped within 24 hours, and the East Coast Warehouse shipped 372 orders within 24 hours. Maureen's null hypothesis is ___.

Are women more risk averse in the stock market? A sample of men and women were asked the following question: "If both the stock market and a stock you owned dropped 25% in three months, would you buy more shares while price is low?" Of 965 women, 338 said yes. Of 1066 men, 554 said yes. At the α = 0.05 level of significance, is there evidence that the proportion of women who would buy more shares while the price is low is less than the proportion of men?

Maxwell Cantor, VP of human resources of Asimov Aerospace Industries Inc., is reviewing the technical certifications of employees in different divisions of the company. His goal is having 90% of technical employees with up-to-date certifications. The north-east division typically has maintained higher rates of up-to-date certifications than the southern division. He selects a random sample of 250 employees from the north-east division and 300 from the southern division and finds out that the number of employees with up-to-date certifications are 230 in the north-east division and 265 in the southern division. Maxwell's alternative hypothesis is ___.

Golf course designer Roberto Langabeer is evaluating two sites, Palmetto Dunes and Ocean Greens, for his next golf course. He wants to prove that Palmetto Dunes residents (population 1) play golf more often than Ocean Greens residents (population 2). Roberto commissions a market survey to test this hypothesis. The market researcher used a random sample of 64 individuals from each suburb, and reported the following: ₁ = 16 times per month and ₂ = 14 times per month. Assume that $\sigma$ ₁ = 4 and $\sigma$ ₂ = 3. With $\alpha$ =.01, the observed z value is ___.

Maureen McIlvoy, owner and CEO of a mail order business for wind surfing equipment and supplies, is reviewing the order filling operations at her warehouses. Her goal is 100% of orders shipped within 24 hours. In previous years, neither warehouse has achieved the goal, but the East Coast Warehouse has consistently out-performed the West Coast Warehouse. Her staff randomly selected 200 orders from the West Coast Warehouse (population 1) and 400 orders from the East Coast Warehouse (population 2), and reports that 190 of the West Coast Orders were shipped within 24 hours, and the East Coast Warehouse shipped 356 orders within 24 hours. Assuming $\alpha$ = 0.05, the observed z value is ___.

You are evaluating investing in a cognitive training company. For this reason, you want to determine whether users who complete at least 75% of the recommended daily training for two months show improved levels of reading comprehension and problem solving skills. You select a random sample of new users and get test scores for each participant in the sample. The test score is a composite of reading comprehension and problem solving skills. Two months later, you randomly select 15 users who have completed 75% or more of the recommended training for the last two months and have them take a test similar to the initial test. You are interested in determining whether the average test score before training is different than the average test score after training for this sample. The after-training average is 92.8, which is 2.7 points higher than the before-training average. The sample standard deviations of the differences is 1.2. You can assume that the differences are normally distributed in the population. The degrees of freedom for this test are ___.

To determine if there is a difference in the speed of a wireless router from two different manufacturers, a network manager will draw independent samples of wireless routers from the two manufacturers and compare the difference in the sample means for the connection speed.

A researcher is estimating the average difference between two population means based on matched?pairs samples. She gathers data on each pair in the study resulting in: Assume that the data are normally distributed in the population. To obtain a 90% confidence interval, the table t value would be ___.

A researcher believes a new diet should improve weight gain. To test his hypothesis a random sample of 10 people on the old diet and an independent random sample of 10 people on the new diet were selected. The selected people on the old diet gain an average of 5 pounds with a standard deviation of 2 pounds, while the average gain for selected people on the new diet was 8 pounds with a standard deviation of 1.5 pounds. Assume that the values are normally distributed in each population and that the population variances are approximately equal. Using = 0.05, the critical t value used from the table for this is ___.

If the populations are normally distributed but the population variances are unknown the t statistic can be used as the basis for statistical inferences about the difference in two population means using two independent random samples.

In conducting a matched-pairs hypothesis test, the null and alternative hypotheses always represent one-tailed tests.

A researcher wants to conduct a before/after study on 11 subjects to determine if a new cholesterol medication results in higher HDL cholesterol readings. The null hypothesis is that the average difference is zero while the alternative hypothesis is that the average difference is $\textbf{ not }$ zero. Scores are obtained on the subjects both before and after taking the medication. After subtracting the after scores from the before scores, the average difference is computed to be -2.40 with a sample standard deviation of 1.21. Assume that the differences are normally distributed in the population. The degrees of freedom for this test are ___.

The difference in two sample means is normally distributed for sample sizes ≥ 30, only if the populations are normally distributed.

You are evaluating investing in a cognitive training company. For this reason, you want to determine whether users who complete at least 75% of the recommended daily training for two months show improved levels of reading comprehension and problem solving skills. You select a random sample of new users and get test scores for each participant in the sample. The test score is a composite of reading comprehension and problem solving skills. Two months later, you randomly select 15 users who have completed 75% or more of the recommended training for the last two months and have them take a test similar to the initial test. You are interested in determining whether the average test score before training is different than the average test score after training for this sample. The after-training average is 92.8, which is 2.7 points higher than the before-training average. The sample standard deviations of the differences is 1.2. You use a significance level of 0.10, and you can assume that the differences are normally distributed in the population. The table t value for this test is ___.

A researcher is conducting a matched?pairs study. She gathers data on each pair in the study resulting in: Assume that the data are normally distributed in the population. The level of significance is selected to be 0.10. If the alternative hypothesis is that the average difference is greater than zero, the null hypothesis would be rejected if the observed value of t is ___.

If the sample sizes are small, but the populations are normally distributed and the population variances are known, the z statistic can be used as the basis for statistical inferences about the difference in two population means using two independent random samples.

If the populations are normally distributed and the variances of the two populations are equal, it is appropriate to use the "pooled" formula to determine the t statistic for the hypothesis test of the difference in the two population means.

You are interested in determining the difference in two population means. You select a random sample of 8 items from the first population and 8 from the second population and then compute a 95% confidence interval. The sample from the first population has an average of 12.2 and a standard deviation of 0.8. The sample from the second population has an average of 11.7 and a standard deviation of 1.0. Assume that the values are normally distributed in each population. The point estimate for the difference in means of these two populations is ___.