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Ch. 8 - Hypothesis Testing with Two Samples
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLarson 8th EditionCh. 8 - Hypothesis Testing with Two SamplesProblema 8.RE.11
Capitolo 8, Problema 8.RE.11

In Exercises 11–16, test the claim about the difference between two population means μ1 and μ2 at the level of significance α. Assume the samples are random and independent, and the populations are normally distributed.


Claim: μ1= μ2; α=0.05. Assume (σ1)^2 = (σ2)^2


Sample statistics: x̅1=228, s1=27, n1= 20 and x̅2=207, s2=25, n2= 13

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Identify the null hypothesis \( H_0 \) and the alternative hypothesis \( H_a \). Since the claim is \( \mu_1 = \mu_2 \), we have \( H_0: \mu_1 = \mu_2 \) and \( H_a: \mu_1 \neq \mu_2 \) because this is a two-tailed test.
Since the population variances are assumed equal (\( \sigma_1^2 = \sigma_2^2 \)), calculate the pooled sample variance \( s_p^2 \) using the formula: \[ s_p^2 = \frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2} \]
Calculate the test statistic \( t \) for the difference between two means using the pooled variance: \[ t = \frac{\bar{x}_1 - \bar{x}_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}} \] where \( s_p = \sqrt{s_p^2} \).
Determine the degrees of freedom for the test, which is \( df = n_1 + n_2 - 2 \).
Find the critical value(s) from the \( t \)-distribution table for a two-tailed test at significance level \( \alpha = 0.05 \) and \( df \) degrees of freedom. Then compare the calculated \( t \)-statistic to the critical value(s) to decide whether to reject or fail to reject the null hypothesis.

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Hypothesis Testing for Two Population Means

This involves testing whether there is a statistically significant difference between the means of two populations. The null hypothesis (H0) typically states that the means are equal (μ1 = μ2), while the alternative hypothesis suggests they are different. The test uses sample data to decide whether to reject H0 at a given significance level (α).
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Difference in Means: Hypothesis Tests

Pooled Variance and Equal Population Variances Assumption

When the population variances are assumed equal (σ1² = σ2²), a pooled variance estimate combines the sample variances to improve the accuracy of the test statistic. This pooled variance is a weighted average of the two sample variances, reflecting the assumption of homogeneity of variance, which affects the calculation of the standard error.
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Level of Significance (α) and Decision Rule

The level of significance, α, is the threshold probability for rejecting the null hypothesis, commonly set at 0.05. It represents the risk of a Type I error (false positive). The test statistic is compared to critical values from the t-distribution, and if it falls in the rejection region, H0 is rejected, indicating a significant difference between means.
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Probability of Non-Mutually Exclusive Events Example
Pratica correlata
Domanda del libro di testo

In Exercises 17 and 18, (c) find the standardized test statistic t, Assume the samples are random and independent, and the populations are normally distributed.


A real estate agent claims that there is no difference between the mean household incomes of two neighborhoods. The mean income of 12 randomly selected households from the first neighborhood is \$52,750 with a standard deviation of \$2900. In the second neighborhood, 10 randomly selected households have a mean income of \$51,200 with a standard deviation of \$2225. At α=0.01, can you reject the real estate agent’s claim? Assume the population variances are equal.

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Domanda del libro di testo

In Exercises 1–4, classify the two samples as independent or dependent and justify your answer.


Sample 1: The fuel efficiencies of 12 cars

Sample 2: The fuel efficiencies of the same 12 cars using an alternative fuel




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Domanda del libro di testo

"In Exercises 17 and 18, (b) find the critical value(s) and identify the rejection region(s), Assume the samples are random and independent, and the populations are normally distributed.


A real estate agent claims that there is no difference between the mean household incomes of two neighborhoods. The mean income of 12 randomly selected households from the first neighborhood is \$52,750 with a standard deviation of \$2900. In the second neighborhood, 10 randomly selected households have a mean income of \$51,200 with a standard deviation of \$2225. At α=0.01, can you reject the real estate agent’s claim? Assume the population variances are equal."

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Domanda del libro di testo

In Exercises 5–8, test the claim about the difference between two population means μ1 and μ2 at the level of significance α. Assume the samples are random and independent, and the populations are normally distributed.


Claim: μ1=μ2; α=0.01


Population statistics: σ1= 52 and σ2= 68


Sample statistics: x̅1 = 5595, n1 = 156, and x̅2= 5575, n2= 216

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Domanda del libro di testo

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


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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Domanda del libro di testo

In Exercises 5–8, test the claim about the difference between two population means μ1 and μ2 at the level of significance α. Assume the samples are random and independent, and the populations are normally distributed.


Claim: μ1<μ2; α=0.10


Population statistics: σ1= 0.11 and σ2= 0.10


Sample statistics: x̅1 = 0.28, n1 = 41, and x̅2= 0.33, n2= 34

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