Question

Megan would like to test whether the average commute time is noticeably different for students at two different high schools in her city, using a significance level of 0.1. In a sample of 61 students at High School 1, she gets a sample mean of 21.2 minutes and a sample standard deviation of 3.3 minutes. In a sample of 61 students at High School 2, she gets a sample mean of 17.7 minutes and a sample standard deviation of 4.1 minutes. Assume the population variances are equal.

What are the null and alternative hypotheses?

Answer 1

Null Hypothesis
`(\(H_0\))`: The average commute time for students at High School 1
`(\(\mu_1\))` is equal to the average commute time for students at High School 2
`(\(\mu_2\))`.

Alternative Hypothesis
`(\(H_1\))`: The average commute time for students at High School 1 (
`\(\mu_1\))` is noticeably different from the average commute time for students at High School 2
`(\(\mu_2\))`.

Step-by-step explanation:

In hypothesis testing, Megan aims to compare the average commute times of students at two different high schools. The null hypothesis
`(\(H_0\))` states that there is no significant difference in average commute times
`(\(\mu_1 = \mu_2\))`, while the alternative hypothesis (
`H_1\))` asserts that there is a noticeable difference in average commute times
`(\(\mu_1 \\eq \mu_2\))`.

To perform the hypothesis test, Megan can use a two-sample t-test since she is comparing means from two independent samples. The formula for the test statistic is given by:

`\[ t = \frac{(\bar{x}_1 - \bar{x}_2)}{\sqrt{\left((s_1^2)/(n_1)\right) + \left((s_2^2)/(n_2)\right)}} \]`

Given the sample means
`(\(\bar{x}_1 = 21.2\)`,
`\(\bar{x}_2 = 17.7\))`, sample standard deviations
`(\(s_1 = 3.3\)`,
`\(s_2 = 4.1\))`, and sample sizes
`(\(n_1 = n_2 = 61\))`, Megan can calculate the test statistic. After obtaining the test statistic, she can compare it to the critical value from the t-distribution at a significance level of 0.1 to make a decision regarding the null hypothesis.

This hypothesis test will help Megan determine whether the observed difference in average commute times is statistically significant or if it could be due to random variation.