To solve this hypothesis test using the p-value method, let’s follow the steps systematically:

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1. State the Hypotheses and Identify the Claim

• Null Hypothesis ($H_0$): $\mu = 140$ (The mean number of Facebook friends is 140.)
• Alternative Hypothesis ($H_a$): $\mu > 140$ (The mean number of Facebook friends is greater than 140.)
  • This is a right-tailed test.
• Claim: The alternative hypothesis ($H_a$) represents the claim.

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2. Given Data

• Population standard deviation: $\sigma = 38.2$
• Sample size: $n = 38$
• Sample mean: $\bar{x} = 145$
• Significance level: $\alpha = 0.10$

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3. Compute the Test Statistic

The test statistic for a population mean with known standard deviation is calculated using the formula for the $z$-score: $z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}$

Substitute the values: $z = \frac{145 - 140}{\frac{38.2}{\sqrt{38}}}$

First, compute the denominator: $\frac{\sigma}{\sqrt{n}} = \frac{38.2}{\sqrt{38}} \approx \frac{38.2}{6.16} \approx 6.2$

Now compute $z$: $z = \frac{145 - 140}{6.2} = \frac{5}{6.2} \approx 0.81$

Thus, the test statistic is: $z = 0.81$

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4. Find the P-value

Using standard normal distribution tables or a calculator:

• The p-value is the area to the right of $z = 0.81$ on the standard normal curve.
• Look up $z = 0.81$ in the z-table: The cumulative probability up to $z = 0.81$ is approximately $0.7910$.
• Therefore: $\text{P-value} = 1 - 0.7910 = 0.209$

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5. Make the Decision

• Compare the p-value ($0.209$) to the significance level ($\alpha = 0.10$):
  • $0.209 > 0.10$: Fail to reject the null hypothesis ($H_0$).

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6. Summarize the Results

At the $0.10$ significance level, there is insufficient evidence to conclude that the mean number of Facebook friends for high school students in this country is greater than 140.

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Follow-Up Questions

1. Why is the test statistic calculated using the population standard deviation instead of the sample standard deviation?
2. What would happen if the significance level $\alpha$ were decreased to 0.05?
3. How does the sample size affect the precision of the test statistic?
4. Can you explain the importance of the p-value in hypothesis testing?
5. What does a right-tailed test imply about the claim being made?

Tip: Always visualize the normal curve and mark the critical region and p-value to gain a better intuition for hypothesis tests.