Question

use the following information to complete steps (a) through (d) below. a random sample of size ( n_1 = 31 ) results in a sample mean of 124.3 and a sample standard deviation of 8.5. an independent sample of size ( n_2 = 50 ) results in a sample mean of 131.8 and sample standard deviation of 7.3. does this constitute sufficient evidence to conclude that the population means differ at the ( alpha = 0.10 ) level of significance? (a) what type of test should be used? a. a hypothesis test regarding two population standard deviations b. a hypothesis test regarding the difference of two means using welchs approximate t. c. a hypothesis test regarding the difference between two population proportions from independent samples d. a hypothesis test regarding the difference of two means using a matched - pairs design (b) determine the null and alternative hypotheses. choose the correct answer below. a. ( h_0:mu_1=mu_2;h_1:mu_1leqmu_2 ) b. ( h_0:mu_1=mu_2;h_1:mu_1gtmu_2 ) c. ( h_0:mu_1leqmu_2;h_1:mu_1
eqmu_2 ) d. ( h_0:mu_1=mu_2;h_1:mu_1
eqmu_2 )

Explanation:

Step1: Analyze the problem type

We are comparing two independent samples to check if the population means differ. Since the variances might not be equal (different sample standard deviations \(s_1 = 8.5\), \(s_2=7.3\)), Welch's approximate \(t -\)test for the difference of two means is appropriate.

Step2: Determine the null and alternative hypotheses

The null hypothesis \(H_0\) assumes no difference between the population means, i.e., \(\mu_1=\mu_2\). The alternative hypothesis \(H_1\) for a two - tailed test (checking if means differ) is \(\mu_1
eq\mu_2\)

Answer:

(a) B. A hypothesis test regarding the difference of two means using Welch's approximate \(t\)
(b) D. \(H_0:\mu_1 = \mu_2;H_1:\mu_1
eq\mu_2\)