To solve this hypothesis test and find the P-value, we need to follow these steps:

Given:

• Sample size ($n$) = 78
• Sample mean ($\bar{x}$) = 278
• Population standard deviation ($\sigma$) = 40
• Hypothesized population mean ($\mu_0$) = 270
• Significance level ($\alpha$) = 0.11

Step 1: Set up the hypotheses.

• Null hypothesis ($H_0$): $\mu = 270$ (The mean score is 270)
• Alternative hypothesis ($H_a$): $\mu > 270$ (The mean score is greater than 270)

Step 2: Calculate the test statistic.

We use the Z-test for the population mean because the population standard deviation is known. The formula for the Z-test statistic is:

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

Where:

• $\bar{x}$ is the sample mean,
• $\mu_0$ is the hypothesized population mean,
• $\sigma$ is the population standard deviation,
• $n$ is the sample size.

Substitute the given values into the formula:

$Z = \frac{278 - 270}{\frac{40}{\sqrt{78}}}$

First, calculate the denominator:

$\frac{40}{\sqrt{78}} = \frac{40}{8.83} \approx 4.53$

Now, calculate the Z-score:

$Z = \frac{278 - 270}{4.53} \approx \frac{8}{4.53} \approx 1.77$

Step 3: Find the P-value.

Since this is a one-tailed test (right-tailed), we can find the P-value by looking up the Z-value in the standard normal distribution table, or using a calculator. The Z-value of 1.77 corresponds to a P-value of approximately 0.0384.

Step 4: Conclusion.

Now that we have the P-value, we can compare it to the significance level $\alpha = 0.11$.

• If the P-value is less than $\alpha$, we reject the null hypothesis.
• If the P-value is greater than $\alpha$, we fail to reject the null hypothesis.

In this case, the P-value (0.0384) is less than the significance level ($\alpha = 0.11$). Therefore, we reject the null hypothesis.

Final Answer:

There is enough evidence to support the administrator's claim that the mean score for the state's eighth graders is greater than 270.

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Would you like a more detailed explanation of any part of the process or further assistance with related concepts?

Here are some related questions for you to explore:

1. How do you perform a Z-test for proportions?
2. What is the difference between a one-tailed and a two-tailed test?
3. How do you calculate the confidence interval for a population mean using the Z-distribution?
4. What are the assumptions for using a Z-test?
5. How does increasing the sample size affect the results of a hypothesis test?

Tip: Always ensure that your sample size is large enough (typically $n > 30$) for the Z-test to be valid when the population standard deviation is known.