Questions: Caden is an editor for a local newspaper. The newspaper's editor-in-chief asked Caden to determine whether the average distance a reader lives from the newspaper's headquarters is greater than 25 miles. He sent a survey to a random selection of the newspaper's readers and received 30 responses. Using the results from the current survey and several previous surveys, Caden decided to assume that the population standard deviation for the distance the readers live from the newspaper's headquarters is 12.3 miles. (a) H0: μ ≤ 25 ; Ha: μ > 25, which is a right-tailed test. (b) z0=0.62, p-value is =0.268 (c) Which of the following are appropriate conclusions for this hypothesis test? Select all that apply.

Caden is an editor for a local newspaper. The newspaper's editor-in-chief asked Caden to determine whether the average distance a reader lives from the newspaper's headquarters is greater than 25 miles. He sent a survey to a random selection of the newspaper's readers and received 30 responses. Using the results from the current survey and several previous surveys, Caden decided to assume that the population standard deviation for the distance the readers live from the newspaper's headquarters is 12.3 miles.
(a) H0: μ ≤ 25 ; Ha: μ > 25, which is a right-tailed test.
(b) z0=0.62, p-value is =0.268
(c) Which of the following are appropriate conclusions for this hypothesis test? Select all that apply.

Transcript text: Caden is an editor for a local newspaper. The newspaper's editor-in-chief asked Caden to determine whether the average distance a reader lives from the newspaper's headquarters is greater than 25 miles. He sent a survey to a random selection of the newspaper's readers and received 30 responses. Using the results from the current survey and several previous surveys, Caden decided to assume that the population standard deviation for the distance the readers live from the newspaper's headquarters is 12.3 miles. (a) $H_{0}: \mu \leq 25 ; H_{a}: \mu>25$, which is a right-tailed test. (b) $z_{0}=0.62, p$-value is $=0.268$ (c) Which of the following are appropriate conclusions for this hypothesis test? Select all that apply.

Solution

Solution Steps

Step 1: Define Hypotheses

We are conducting a hypothesis test to determine if the average distance a reader lives from the newspaper's headquarters is greater than 25 miles. The hypotheses are defined as follows:

Null Hypothesis ($H_0$): $ \mu \leq 25 $
Alternative Hypothesis ($H_a$): $ \mu > 25 $

Step 2: Calculate Standard Error

The standard error ($SE$) is calculated using the formula:

\[ SE = \frac{\sigma}{\sqrt{n}} = \frac{12.3}{\sqrt{30}} \approx 2.2457 \]

Step 3: Calculate Test Statistic

The test statistic ($Z$) is calculated using the formula:

\[ Z = \frac{\bar{x} - \mu_0}{SE} = \frac{43 - 25}{2.2457} \approx 8.0155 \]

Step 4: Calculate P-value

For a right-tailed test, the p-value is calculated as:

\[ P = 1 - T(z) \approx 0.0 \]

Step 5: Decision Rule

We compare the p-value to the significance level ($\alpha = 0.05$):