Question

a researcher collated data on americans’ leisure time activities. she found the mean number of hours spent watching television each weekday to be 2.7 hours with a standard deviation of 0.2 hours. jonathan believes that his football team buddies watch less television than the average american. he gathered data from 40 football teammates and found the mean to be 2.3. which of the following are the correct null and alternate hypotheses?

• $h_0: \mu = 2.7, h_a: \mu < 2.7$
• $h_0: \mu \

eq 2.7, h_a: \mu = 2.3$

• $h_0: \mu = 2.7, h_a: \mu \

eq 2.7$

• $h_0: \mu < 2.7, h_a: \mu \geq 2.3$

Explanation:

Step1: Recall Hypothesis Testing Basics

In hypothesis testing, the null hypothesis (\(H_0\)) is a statement of no difference or the status quo. The alternative hypothesis (\(H_a\)) is what we are trying to find evidence for. Here, the population mean (Americans' mean TV - watching hours on weekdays) is \(\mu = 2.7\) hours. Jonathan believes his football team buddies watch less TV than the average American.

Step2: Formulate Null Hypothesis

The null hypothesis should be about the population mean being equal to the known average. So, \(H_0:\mu = 2.7\).

Step3: Formulate Alternative Hypothesis

Since Jonathan believes his football team buddies (a sample from a population) have a mean TV - watching time less than the average American, the alternative hypothesis is \(H_a:\mu<2.7\) (because we are testing if the mean of his group is less than the population mean of 2.7).

Let's analyze the other options:

• Option 2: The alternative hypothesis should be about the population mean (\(\mu\)) relative to 2.7, not equal to 2.3 (2.3 is the sample mean). So this is incorrect.
• Option 3: The alternative hypothesis here is a two - tailed test (\(\mu

eq2.7\)), but we have a one - tailed test (less than) situation. So this is incorrect.

• Option 4: The null hypothesis is stated as \(\mu < 2.7\) which is not the status quo (the status quo is \(\mu = 2.7\)). So this is incorrect.

Answer:

\(H_0:\mu = 2.7,H_a:\mu < 2.7\) (the first option)