Question

a commonly cited standard for one - way length (duration) of school bus rides for elementary school children is 30 minutes. a local government office in a rural area randomly samples 100 elementary school children in their district and find an average one - way commute time of 38 minutes with a standard deviation of 8 minutes. which of the following is the correct set of hypotheses for testing if the average commute time of elementary school students in this district is different than the commonly cited standard of 30 minutes?

$h_0:x = 38;h_a:mu=30$

$h_0:mu = 30;h_a:mu>30$

$h_0:mu = 30;h_a:mu
eq30$

$h_0:mu = 30;h_a:x = 38$

$h_a:mu = 30;h_0:x
eq30$

Explanation:

Step1: Understand null and alternative hypotheses

The null hypothesis \(H_0\) is a statement of no difference or equality. The alternative hypothesis \(H_A\) is what we are trying to find evidence for.
We are testing if the average (\(\mu\)) is different from 30.

Step2: Analyze each option

• Option 1: \(H_0:x = 38;H_A:\mu=30\) is incorrect. The null hypothesis should be about the population parameter (\(\mu\)), not the sample statistic (\(x\)).
• Option 2: \(H_0:\mu = 30;H_A:\mu>30\) is a one - sided test. But we want to test if it is different (two - sided), not just greater.
• Option 3: \(H_0:\mu = 30;H_A:\mu

eq30\) is correct. The null hypothesis is that the population mean \(\mu\) is equal to the standard value (30), and the alternative is that it is not equal (two - sided test for difference).

• Option 4: \(H_0:\mu = 30;H_A:x = 38\) is incorrect. The alternative hypothesis should be about the population parameter (\(\mu\)), not the sample statistic (\(x\)).
• Option 5: \(H_A:\mu = 30;H_0:x

eq30\) has the null and alternative hypotheses reversed. The null hypothesis should be the statement of no difference (equality) about the population parameter.

Answer:

\(H_0:\mu = 30;H_A:\mu
eq30\) (the third option)