Exam 14: Hypothesis Testing: Comparing Two Populations

Two independent samples of sizes 30 and 40 are randomly selected from two populations to test the difference between the population means, µ $\mu$ ₂ - µ $\mu$ _1, where the population variances are unknown. Which of the following best describes the sampling distribution of the sample mean difference $\bar { x } _ { 1 } - \bar { x } _ { 2 }$ ? A. Normal. B. t -distributed with 68 degrees of freedom. C. Approximately normal. D. None of these choices are correct.

A management consultant wants to compare the incomes of graduates of MBA programs with those of graduates with Bachelor's degrees. In a random sample of the incomes of 20 people taken five years after they received their MBAs, the consultant found the mean salary and the standard deviation to be $45 300 and $9600, respectively. A random sample of the incomes of 25 people taken five years after they received their Bachelor's degrees yielded a mean salary of $43 600 with a standard deviation of $12 300. Can we infer at the 10% level of significance that the population means differ?

Two independent samples of sizes 30 and 35 are randomly selected from two normal populations with equal variances. Which of the following is the test statistic that should be used to test the difference between the population means? A. Student $t$ with 64 degrees of freedom. B. Chi-square C. Z D. Student $t$ with 63 degrees of freedom.

In testing the hypotheses: H₀: $\mu$ ₁ - $\mu$ ₂ = 0 H_A: $\mu$ ₁ - $\mu$ ₂ $\neq$ 0, two random samples from two normal populations produced the following statistics: n₁ = 51, x₁-bar = 35, s₁ = 28. n₂ = 40, x₂-bar = 28, s₂ = 10. Assume that the two population variances are different. a. Estimate with 95% confidence the difference between the two population means. b. Explain how to use this confidence interval for testing the hypotheses.

In testing the hypotheses: H₀: p₁ - p₂ = 0.10 H₁ : p₁ - p₂ $\neq$ 0.10 we find the following statistics: n₁ = 150, x₁ = 72. n₂ = 175, x₂ = 70. a. What is the p-value of the test? b. Briefly explain how to use the p-value to test the hypotheses, at a 1% level of significance.

The managing director of a breakfast cereal manufacturer claims that families in which both spouses work are much more likely to be consumers of his product than those with only one working spouse. To prove his point, he commissions a survey of 300 families in which both spouses work and 300 families with only one working spouse. Each family is asked whether the company's cereal is eaten for breakfast. The results are shown below. Two spouses working One spouse working Eat cereal 114 87 Don't eat cereal 186 213 Use the p-value method to test the managing director's claim, at a 5% significance level of significance.

In testing the hypotheses: H₀: p₁ - p₂ = 0.10 H_A: p₁ - p₂ > 0.10, we found the following statistics: n₁ = 350, x₁ = 178. n₂ = 250, x₂ = 112. a. What is the p-value of the test? b. Use the p-value to test the hypotheses at the 10% level of significance. c. Estimate with 90% confidence the difference between the two population proportions.

A politician regularly polls her electorate to ascertain her level of support among voters. This month, 652 out of 1158 voters support her. Five months ago, 412 out of 982 voters supported her. At the 1% significance level, can she claim that support has increased by at least 10 percentage points?

We can design a matched pairs experiment when the data collected are: A. observational. B. experimental. C. controlled. D. All these choices are correct.

Thirty-five employees who completed two years of tertiary education were asked to take a basic mathematics test. The mean and standard deviation of their marks were 75.1 and 12.8, respectively. In a random sample of 50 employees who only completed high school, the mean and standard deviation of the test marks were 72.1 and 14.6, respectively. a. Estimate with 90% confidence the difference in mean scores between the two groups of employees. b. Explain how to use the interval estimate in part a. to test the hypotheses.

Thirty-five employees who completed two years of tertiary education were asked to take a basic mathematics test. The mean and standard deviation of their marks were 75.1 and 12.8, respectively. In a random sample of 50 employees who only completed high school, the mean and standard deviation of the test marks were 72.1 and 14.6, respectively. Can we infer at the 10% significance level that a difference exists between the two groups?

In testing the difference between two population means, using two independent samples, we use the pooled variance in estimating the standard error of the sampling distribution of the sample mean difference $\bar { x } _ { 1 } - \bar { x } _ { 2 }$ if the: A. sample sizes are both large. B. populations are normally distributed. C. populations have equal variances. D. populations are normal with equal variances.

The pooled proportion estimate is used to estimate the standard error of the difference between two proportions when the proportions of two populations are hypothesized to be equal.

A marketing consultant is studying the perceptions of married couples concerning their weekly food expenditures. He believes that the husband's perception would be higher than the wife's. To judge his belief, he takes a random sample of 10 married couples and asks each spouse to estimate the family food expenditure (in dollars) during the previous week. The data are shown below. Couple Husband Wife 1 380 270 2 280 300 3 215 185 4 350 320 5 210 180 6 410 390 7 250 250 8 360 320 9 180 170 10 400 330 Can the consultant conclude at the 5% significance level that the husband's estimate is higher than the wife's estimate?

Which of the following is a required condition for using the normal approximation to the binomial distribution in testing the difference between two population proportions? A. >5 and >5 . B. 1- >5. C. >5, 1- >5,>5 and 1- >5 . D. 1- >5 and 1- >5

In testing the null hypothesis H₀ = $p _ { 1 } - p _ { 2 }$ = 0, if H₀ is true, the test could lead to: A. a Type I error. B. a Type II error. C. either a Type I or a Type II error. D. None of these choices are correct.

When the necessary conditions are met, a two-tail test is being conducted to test the difference between two population proportions. The two sample proportions are p₁ = 0.25 and p₂ = 0.20 and the standard error of the sampling distribution of $p _ { 1 } - p _ { 2 }$ is 0.04. The calculated value of the test statistic will be: A. z=0.25 B. z=1.25 C. t=0.25 D. t=0.80

Test the following hypotheses at the 5% level of significance: H₀: µ $\mu$ ₁ - µ $\mu$ ₂ = 0 H_A: µ $\mu$ ₁ - µ $\mu$ ₂ < 0, given the following statistics: n₁ = 10, x₁ = 58.6, s₁ = 13.45. n₂ = 10, x₂ = 64.6, s₂ = 11.15. Estimate with 95% confidence the difference between the two population means.

In testing the hypotheses: H₀: $\mu$ _D = 5 H_A: $\mu$ _D $\neq$ 5, two random samples from two normal populations produced the following statistics: n_D = 36, x_D = 7.8, s_D = 7.5. What conclusion can we draw at the 5% significance level?

In testing whether the means of two normal populations are equal, summary statistics computed for two independent samples are as follows: n₁ = 25, $\bar { x } _ { 1 }$ = 7.30, s₁ = 1.05. N₂ = 30, $\bar { x } _ { 2 }$ = 6.80, s₂ = 1.20. Assume that the population variances are equal. Then the standard error of the sampling distribution of the sample mean difference $\bar { x } _ { 1 } - \bar { x } _ { 2 }$ is equal to: A. 0.3072. B. 0.0917. C. 0.3028 D. 0.0944.