To analyze the question in your image:

The problem asks us to perform a hypothesis test regarding the mean time spent on Facebook per visit.

(a) State the appropriate null and alternative hypotheses:

Let the population mean time spent on Facebook per visit be $\mu$.

• The null hypothesis ($H_0$) is that the mean time has not increased from the previous value of $20.2$ minutes: $H_0: \mu = 20.2$

• The alternative hypothesis ($H_1$) is that the mean time has increased: $H_1: \mu > 20.2$

This is a one-tailed test because we are testing for an increase in the mean.

(b) Compute the p-value:

Given Information:

• Previous mean ($\mu_0$) = $20.2$
• Sample mean ($\bar{x}$) = $21.2$
• Population standard deviation ($\sigma$) = $4.0$
• Sample size ($n$) = $107$
• Level of significance ($\alpha$) = $0.05$

Step 1: Calculate the test statistic ($z$):

The formula for $z$ is: $z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}}$

Substitute the values: $z = \frac{21.2 - 20.2}{\frac{4.0}{\sqrt{107}}} = \frac{1.0}{\frac{4.0}{10.344}} = \frac{1.0}{0.3869} \approx 2.58$

Step 2: Find the p-value:

For a $z$-score of $2.58$, the p-value is the area to the right of $z$. Using a $z$-table or a calculator: $p \text{-value} \approx 0.0049$

(c) State a conclusion:

Compare the p-value ($0.0049$) to the significance level ($\alpha = 0.05$):

• Since $p \text{-value} < \alpha$, we reject the null hypothesis.

Conclusion: There is sufficient evidence to suggest that the mean time spent on Facebook per visit has increased.

Would you like further clarification or step-by-step instructions on any part?

Five Related Questions:

1. How do you interpret the significance level ($\alpha$) in hypothesis testing?
2. What is the difference between a one-tailed and a two-tailed test?
3. How does sample size impact the $z$-score in a hypothesis test?
4. Can a p-value alone determine the practical significance of results?
5. How can we perform the same test using a TI-84 calculator?

Tip:

When solving hypothesis testing problems, always visualize the distribution curve to understand whether you're working with a one-tailed or two-tailed test.