Exam 14: Hypothesis Testing: Comparing Two Populations

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?

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

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. What conclusion can we draw at the 5% significance level?

If some natural relationship exists between each pair of observations that provides a logical reason to compare the first observation of sample 1 with the first observation of sample 2, the second observation of sample 1 with the second observation of sample 2, and so on, the samples are referred to as:

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.

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 of size 150 from population 1 has 40 successes. A sample of size 250 from population 2 has 30 successes. The value of the test statistic for testing the null hypothesis that the proportion of successes in population 1 exceeds the proportion of successes in population 2 by 0.05 is:

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.

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 political analyst in Perth surveys a random sample of Labor Party members and compares the results with those obtained from a random sample of Liberal Party members. This would be an example of:

A politician has commissioned a survey of blue-collar and white-collar employees in her electorate. The survey reveals that 286 out of 542 blue-collar workers intend to vote for her in the next election, whereas 428 out of 955 white-collar workers intend to vote for her. a. Can she infer at the 5% level of significance that the level of support differs between the two groups of workers? b. What is the p-value of the test? Explain how to use it to test the hypotheses.

The following data were generated from a matched pairs experiment: Pair: 1 2 3 4 5 6 7 Sample 1: 8 15 7 9 10 13 11 Sample 2: 12 18 8 9 12 11 10 a. Estimate with 90% confidence the mean difference. b. Briefly describe what the interval estimate in part a. tells you, and explain how to use it to test 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.

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

The marketing manager of a pharmaceutical company believes that more females than males use its acne medicine. In a recent survey, 2500 teenagers were asked whether or not they use that particular product. The responses, categorised by gender, are summarised below. Use acne medicine Don't use acne medicine Female 540 810 Male 391 759 a. Do these data provide enough evidence at the 10% significance level to support the manager's claim? b. Estimate with 90% confidence the difference in the proportions of male and female users of the acne medicine. c. Describe what the interval estimate in part b. tells you.

The owner of a service station wants to determine whether the owners of new cars (two years old or less) change their cars' oil more frequently than owners of older cars (more than two years old). From his records, he takes a random sample of 10 new cars and 10 older cars and determines the number of times the oil was changed for each in the last 12 months. The data are shown below. Frequency of oil changes in the past 12 months New car owners Old car owners 6 4 3 2 3 1 3 2 4 3 3 2 6 2 5 3 5 2 4 1 Do these data allow the service station owner to infer at the 10% significance level that new car owners change their cars' oil more frequently than older car owners?

Because of the rising costs of industrial accidents, many chemical, mining and manufacturing firms have instituted safety courses. Employees are encouraged to take these courses, which are designed to heighten safety awareness. A company is trying to decide which one of two courses to institute. To help make a decision, eight employees take course 1 and another eight take course 2. Each employee takes a test, which is graded out of a possible 25. The safety test results are shown below. Course 1 14 21 17 14 17 19 20 16 Course 2 20 18 22 15 23 21 19 15 Assume that the scores are normally distributed. Does the data provide sufficient evidence at the 5% level of significance to infer that the marks from course 1 are lower than those from course 2?

In testing the difference between two population means, using two independent samples, the sampling distribution of the sample mean difference $\bar { x } _ { 1 } - \bar { x } _ { 2 }$ is normal if the:

A simple random sample of ten firms was asked how much money (in thousands of dollars) they spent on employee training programs this year and how much they plan to spend on these programs next year. The data are shown below. Firm 1 2 3 4 5 6 7 8 9 10 This year 25 31 12 15 21 36 18 5 9 17 Next year 21 30 18 20 22 36 20 10 8 15 Assume that the populations of amount spent on employee training programs are normally distributed. Can we infer at the 5% significance level that more money will be spent on employee training programs next year than this year?

In testing the null hypothesis H₀ = $p _ { 1 } - p _ { 2 }$ = 0, if H₀ is true, the test could lead to: