Ch. 8 - Hypothesis Testing with Two Samples

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 8 - Hypothesis Testing with Two Samples

문제 8.TM.4

8장, 문제 8.TM.4

In Exercises 4 and 5, use technology to perform a two-sample t-test to determine whether there is a difference in the mint dates and in the values of coins found on a street from 1985 through 1996 for the two mint locations. Write your conclusion as a sentence. Use α = 0.05.

Mint dates of coins (years)

Philadelphia: x̅1=1984.8, s1=8.6

Denver: x̅2=1983.4, s2=8.4

Assume population variances are equal.

검증된 단계별 안내

Identify the null hypothesis $ H_0 $ and the alternative hypothesis $ H_a $. Here, $ H_0: \mu_1 = \mu_2 $ (no difference in mean mint dates) and $ H_a: \mu_1 \neq \mu_2 $ (there is a difference in mean mint dates).

Since the population variances are assumed equal, use the pooled two-sample t-test formula. Calculate the pooled standard deviation $ s_p $ using the formula: \[ s_p = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}} \] Note: You will need the sample sizes $ n_1 $ and $ n_2 $ for Philadelphia and Denver respectively.

Calculate the test statistic $ t $ using the formula: \[ t = \frac{\bar{x}_1 - \bar{x}_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}} \]

Determine the degrees of freedom $ df = n_1 + n_2 - 2 $ and use a t-distribution table or technology to find the critical t-value(s) for a two-tailed test at significance level $ \alpha = 0.05 $.

Compare the calculated $ t $-statistic to the critical value(s). If $ |t| $ is greater than the critical value, reject $ H_0 $; otherwise, fail to reject $ H_0 $. Then, write a conclusion in context, stating whether there is sufficient evidence to conclude a difference in mean mint dates between the two mint locations.

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주요 개념

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Two-Sample t-Test

A two-sample t-test compares the means of two independent groups to determine if there is a statistically significant difference between them. It uses sample means, variances, and sizes to calculate a t-statistic, which is then compared to a critical value based on the chosen significance level (α). This test assumes the data are approximately normally distributed.

Equal Population Variances Assumption

Assuming equal population variances means the variability in both groups is similar, allowing the use of a pooled variance estimate in the t-test. This assumption simplifies calculations and affects the degrees of freedom used. If variances differ greatly, a different test (Welch’s t-test) is more appropriate.

Significance Level (α) and Hypothesis Testing

The significance level α (here 0.05) defines the threshold for rejecting the null hypothesis, which states there is no difference between group means. If the p-value from the t-test is less than α, we reject the null, concluding a significant difference exists. Otherwise, we fail to reject it, indicating insufficient evidence.

관련 실천

교과서 질문

Take this test as you would take a test in class.For each exercise, perform the steps below.

b.Determine whether the hypothesis test is left-tailed, right-tailed, or two-tailed, and whether to use a z-test or a t-test. Explain your reasoning.

A real estate agency says that the mean home sales price in Olathe, Kansas, is greater than in Rolla, Missouri. The mean home sales price for 39 homes in Olathe is \$392,453. Assume the population standard deviation is \$224,902. The mean home sales price for 38 homes in Rolla is \$285,787. Assume the population standard deviation is \$330,578. At α=0.05, is there enough evidence to support the agency’s claim? (Adapted from Realtor.com)

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교과서 질문

Find the critical value(s) for the alternative hypothesis, level of significance , and sample sizes and . Assume that the samples are random and independent, the populations are normally distributed, and the population variances are (a) equal

Ha:μ1≠μ2 , α=0.10 , n1=11 , n2=14

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교과서 질문

Take this test as you would take a test in class.For each exercise, perform the steps below.

e. Decide whether to reject or fail to reject the null hypothesis.

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교과서 질문

Testing the Difference Between Two Means (a) identify the claim and state Ho and Ha .Assume the samples are random and dependent, and the populations are normally distributed.

[APPLET] Passing Play Percentages The passing play percentages of 10 randomly selected NCAA Division 1A college football teams for home and away games in the 2020–2021 season are shown in the table. At , α=0.20 is there enough evidence to support the claim that passing play percentage is different for home and away games? (Source: TeamRankings)

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교과서 질문

Take this test as you would take a test in class.For each exercise, perform the steps below.

a. Identify the claim and state and

b.Determine whether the hypothesis test is left-tailed, right-tailed, or two-tailed, and whether to use a z-test or a t-test. Explain your reasoning.

c.Find the critical value(s) and identify the rejection region(s).

d. Find the appropriate standardized test statistic.

e. Decide whether to reject or fail to reject the null hypothesis.

f. Interpret the decision in the context of the original claim.

A demographics researcher claims that the mean household income in a recent year is different for native-born households and foreign-born households. A sample of 18 native-born households has a mean household income of \$69,474 and a standard deviation of $21,249. A sample of 21 foreign-born households has a mean household income of $64,900 and a standard deviation of \$17,896. At α=0.01, can you support the demographics researcher’s claim? Assume the populations are normally distributed and the population variances are not equal. (Adapted from U.S. Census Bureau)

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교과서 질문

Confounding Variables A pharmaceutical company has applied for approval to market a new arthritis medication. The research involved a test group that was given the medication and another test group that was given a placebo. Describe some possible confounding variables that could influence the results of the study.

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