Exam 14: Hypothesis Testing: Comparing Two Populations

A course coordinator at a university wants to investigate if there is a significant difference in the average final mark of students taking the same university subject in semester 1 or semester 2. A random sample of 30 students is taken from semester 1, with the average final mark is found to be 60% and the standard deviation is 5%. A random sample of 50 students is taken from semester 2, with the average final mark is 57% and the standard deviation is 4%. Assuming that the population variances are equal, is there significant evidence that the population average final mark in this course differs between semester 1 and semester 2. Test at the 5% level of significance.

When the necessary conditions are met, a two-tail test is being conducted to test the difference between two population proportions. If the value of the test statistic z is 2.05, then the p-value is: A. 0.4798. B. 0.0404 C. 0.2399 D. 0.0202.

Which of the following best describes the symbol $\bar { x } _ { \mathrm { D } }$ ? A. The difference in the means of two dependent populations. B. The difference in the means of two independent populations. C. The mean of the differences in the pairs of observations taken from two dependent samples. D. The matched pairs differences.

In testing the difference between the means of two normally distributed populations, the number of degrees of freedom associated with the unequal-variances t-test statistic usually results in a non-integer number. It is recommended that you: A. round up to the nearest integer. B. round down to the nearest integer. C. change the sample sizes until the number of degrees of freedom becomes an integer. D. assume that the population variances are equal, and then use $d f=n_{1}+n_{2}-2$ .

When testing the difference between two population proportions, a t test may be used.

In testing the hypotheses: H₀: p₁ - p₂ = 0 H₁: p₁ - p₂ > 0, we find the following statistics: n₁ = 200, x₁ = 80. n₂ = 200, x₂ = 140. a. What is the p-value of the test? b. What is the conclusion if tested at a 5% significance level? c. Estimate with 95% confidence the difference between the two population proportions.

In testing the hypotheses H₀: p₁ - p₂ = 0 H_A: p₁ - p₂ ≠ 0, we find the following statistics: n₁ = 400, x₁ = 105. n₂ = 500, x₂ = 140. What conclusion can we draw at the 10% significance level?

In testing the hypotheses H₀: p₁ - p₂ = 0 H_A: p₁ - p₂ < 0, we find the following statistics: n₁ = 400, x₁ = 105. n₂ = 500, x₂ = 140. Estimate with 90% confidence the difference between the two population proportions.

The test statistic to test the difference between two population proportions is the Z test statistic, which requires that the sample sizes are each sufficiently large.

A quality control inspector keeps a tally sheet of the numbers of acceptable and unacceptable products that come off two different production lines. The completed sheet is shown below. Production line Acceptable products Unacceptable products 1 152 48 2 136 54 a. What is the p-value of the test? b. Estimate with 95% confidence the difference in population proportions.

In random samples of 25 and 22 from each of two normal populations, we find the following statistics: x₁-bar = 56, s₁ = 8. x₂-bar = 62, s₂ = 8.5. Assume that the population variances are equal. Estimate with 95% confidence the difference between the two population means.

The managing director of a breakfast cereal manufacturer believes 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 Do these data provide enough evidence at the 1% significance level to infer that the proportion of families with two working spouses who eat the cereal is at least 5% larger than the proportion of families with one working spouse who eats the cereal?

A psychologist has performed the following experiment. For each of 10 sets of identical twins who were born 30 years ago, she recorded their annual incomes according to which twin was born first. The results (in $000) are shown below. Twin set First born Second born 1 32 44 2 36 43 3 21 28 4 30 39 5 49 51 6 27 25 7 39 32 8 38 42 9 56 64 10 44 44 Can she infer at the 5% significance level that there is a difference in income between the twins?

A political poll taken immediately prior to a state election reveals that 158 out of 250 male voters and 105 out of 200 female voters intend to vote for the Independent candidate. a. What is the p-value of the test? b. Estimate with 95% confidence the difference between the proportions of male and female voters who intend to vote for the Independent candidate. c. Explain how to use the interval estimate in part b. to test the hypotheses.

Ten functionally illiterate adults were given an experimental one-week crash course in reading. Each of the 10 was given a reading test prior to the course and another test after the course. The results are shown below. Adult 1 2 3 4 5 6 7 8 9 10 Score after course 48 42 43 34 50 30 43 38 41 3 Score before course 31 34 18 30 44 28 34 33 27 32 a. Estimate the mean improvement with 95% confidence. b. Briefly describe what the interval estimate in part a. tells you.

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. Estimate with 95% confidence the difference between the two population proportions. b. Explain how to use the confidence interval in part a. to test the hypotheses.

A quality control inspector keeps a tally sheet of the numbers of acceptable and unacceptable products that come off two different production lines. The completed sheet is shown below. Production line Acceptable products Unacceptable products 1 152 48 2 136 54 Can the inspector infer at the 5% significance level that production line 1 is doing a better job than production line 2?

When the necessary conditions are met, a two-tail test is being conducted to test the difference between two population means, but your statistical software provides only a one-tail area of 0.028 as part of its output. The p-value for this test will be: A. 0.972. B. 0.05. C. 0.056. D. 0.014.

The degrees of freedom for a t test of the difference of population means from two independent samples are n₁ + n₂ - 2.

In order to test the hypotheses: H₀: µ $\mu$ ₁ - µ $\mu$ ₂ = 0 H₁: µ $\mu$ ₁ - µ $\mu$ ₂ $\neq$ 0, we independently draw a random sample of 18 observations from a normal population with standard deviation of 15, and another random sample of 12 from a second normal population with standard deviation of 25. a. If we set the level of significance at 5%, determine the power of the test when ₁ - ₂ = 5. b. Describe the effect of reducing the level of significance on the power of the test.