Ch. 9 - Inferences from Two Samples

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook

Triola 14th Edition

Ch. 9 - Inferences from Two Samples

Problem 9.Q.6

Chapter 9, Problem 9.Q.6

Variation Find the value of the test statistic used for testing the claim that the two samples from Exercise 5 are from populations having the same variation.

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Identify the two sample variances, denoted as \(s_1^2\) and \(s_2^2\), from the data given in Exercise 5. These represent the sample estimates of the population variances.

Determine the sample sizes for each group, denoted as \(n_1\) and \(n_2\), which are the number of observations in each sample.

Formulate the null hypothesis \(H_0\): the population variances are equal, i.e., \(\sigma_1^2 = \sigma_2^2\), and the alternative hypothesis \(H_a\): the population variances are not equal, i.e., \(\sigma_1^2 \neq \sigma_2^2\).

Calculate the test statistic for comparing two variances using the F-test formula: \[ F = \frac{s_1^2}{s_2^2} \] where \(s_1^2\) is the larger sample variance to ensure \(F \geq 1\).

Determine the degrees of freedom for the numerator and denominator as \(df_1 = n_1 - 1\) and \(df_2 = n_2 - 1\), respectively, which will be used to find critical values or p-values from the F-distribution.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Test Statistic for Comparing Variances

When testing if two populations have the same variance, the test statistic commonly used is the F-ratio, which is the ratio of the two sample variances. This statistic follows an F-distribution under the null hypothesis that the population variances are equal.

F-Distribution

The F-distribution is a probability distribution used to compare two variances. It is defined by two degrees of freedom parameters, corresponding to the sample sizes minus one for each sample. The shape of the distribution depends on these degrees of freedom.

Null and Alternative Hypotheses in Variance Testing

In testing for equal variances, the null hypothesis states that the population variances are equal, while the alternative hypothesis states they are not equal (two-tailed) or one variance is greater (one-tailed). The test statistic is compared to critical values from the F-distribution to decide whether to reject the null.

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Textbook Question

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Textbook Question

In Exercises 1–10, based on the nature of the given data, do the following:

a. Pose a key question that is relevant to the given data.

b. Identify a procedure or tool from this chapter or the preceding chapters to address the key question from part (a).

c. Analyze the data and state a conclusion.

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Textbook Question

In Exercises 1–10, based on the nature of the given data, do the following:

a. Pose a key question that is relevant to the given data.

b. Identify a procedure or tool from this chapter or the preceding chapters to address the key question from part (a).

c. Analyze the data and state a conclusion.

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Textbook Question

In Exercises 1–10, based on the nature of the given data, do the following:

a. Pose a key question that is relevant to the given data.

b. Identify a procedure or tool from this chapter or the preceding chapters to address the key question from part (a).

c. Analyze the data and state a conclusion.

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