Let’s break down and solve the hypothesis testing problem step by step.

Given Data:

• Population mean ($\mu_0$): 554
• Sample mean ($\bar{x}$): 567
• Population standard deviation ($\sigma$): 39
• Sample size ($n$): 60
• Significance level ($\alpha$): 0.01

We need to perform a one-tailed test to determine if the population mean SAT score for the preparation course graduates is greater than 554.

(a) Null and Alternative Hypotheses:

$H_0: \mu = 554 \quad (\text{The mean SAT score is not greater than 554})$ $H_1: \mu > 554 \quad (\text{The mean SAT score is greater than 554})$

(b) Test Statistic Type:

Since the population standard deviation ($\sigma$) is known, we use the z-test.

(c) Calculate the Test Statistic:

The formula for the z-test statistic is:

$z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}$

Substitute the given values:

$z = \frac{567 - 554}{39 / \sqrt{60}}$

First, calculate the standard error ($\text{SE}$):

$\text{SE} = \frac{\sigma}{\sqrt{n}} = \frac{39}{\sqrt{60}} \approx 5.032$

Now, compute $z$:

$z = \frac{567 - 554}{5.032} \approx \frac{13}{5.032} \approx 2.584$

So, $z \approx 2.584$.

(d) Find the p-value:

For a one-tailed test, the p-value corresponds to the area to the right of $z = 2.584$ in the standard normal distribution.

Using a z-table or statistical software, the p-value for $z = 2.584$ is approximately:

$p \approx 0.0049$

(e) Decision:

Compare the p-value to the significance level ($\alpha = 0.01$):

• $p = 0.0049 < 0.01$, so we reject the null hypothesis.

Conclusion:

Yes, we can support the preparation course’s claim that the population mean SAT score of its graduates is greater than 554.

Would you like further explanation of any step or need additional help with hypothesis testing concepts?

Related Questions:

1. What is the role of a significance level in hypothesis testing?
2. How does a one-tailed test differ from a two-tailed test?
3. What assumptions are made when using the z-test?
4. How would the result differ if the population standard deviation was unknown?
5. Can you explain how to calculate a confidence interval for the mean SAT score?

Tip:

Always verify whether the sample size is large enough or the population follows a normal distribution to ensure the validity of the z-test.