Questions: A random sample of 84 eighth grade students' scores on a national mathematics assessment test has a mean score of 288. This test result prompts a state school administrator to declare that the mean score for the state's eighth graders on this exam is more than 280. Assume that the population standard deviation is 36. At α=0.08, is there enough evidence to support the administrator's claim? Complete parts (a) through (e) (a) Write the claim mathematically and identify H0 and Ha. Choose the correct answer below. A. H0: μ=280 (claim) B. H0: μ<280 Ha: μ>280 Ha: μ ≥ 280 (claim) C. H0: μ=280 D. H0: μ ≥ 280 (claim) E. H0: μ ≤ 280 (claim) Ha: μ>280 Ha: μ>280 (claim) Ha: μ<280 F. H0: μ ≤ 280 Ha: μ>280 (claim)

A random sample of 84 eighth grade students' scores on a national mathematics assessment test has a mean score of 288. This test result prompts a state school administrator to declare that the mean score for the state's eighth graders on this exam is more than 280. Assume that the population standard deviation is 36. At α=0.08, is there enough evidence to support the administrator's claim? Complete parts (a) through (e) (a) Write the claim mathematically and identify H0 and Ha. Choose the correct answer below. A. H0: μ=280 (claim) B. H0: μ<280 Ha: μ>280 Ha: μ ≥ 280 (claim) C. H0: μ=280 D. H0: μ ≥ 280 (claim) E. H0: μ ≤ 280 (claim) Ha: μ>280 Ha: μ>280 (claim) Ha: μ<280 F. H0: μ ≤ 280 Ha: μ>280 (claim)
Transcript text: A random sample of 84 eighth grade students' scores on a national mathematics assessment test has a mean score of 288. This test result prompts a state school administrator to declare that the mean score for the state's eighth graders on this exam is more than 280. Assume that the population standard deviation is 36. At $\alpha=0.08$, is there enough evidence to support the administrator's claim? Complete parts (a) through (e) (a) Write the claim mathematically and identify $\mathrm{H}_{0}$ and $\mathrm{H}_{\mathrm{a}}$. Choose the correct answer below. A. $H_{0}: \mu=280$ (claim) B. $\mathrm{H}_{0}: \mu<280$ $\mathrm{H}_{\mathrm{a}}: \mu>280$ $H_{a}: \mu \geq 280$ (claim) C. $H_{0}: \mu=280$ D. $\mathrm{H}_{0}: \mu \geq 280$ (claim) E. \[ \begin{array}{l} H_{0}: \mu \leq 280 \text { (claim) } \\ H_{a}: \mu>280 \end{array} \] $H_{a}: \mu>280$ (claim) $\mathrm{H}_{\mathrm{a}}: \mu<280$ F. $H_{0}: \mu \leq 280$ $H_{a}: \mu>280$ (claim)
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Solution

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Solution Steps

To determine if there is enough evidence to support the administrator's claim that the mean score for the state's eighth graders on the exam is more than 280, we will perform a hypothesis test for the population mean. Here are the steps:

  1. State the hypotheses:

    • Null hypothesis (\(H_0\)): \(\mu = 280\)
    • Alternative hypothesis (\(H_a\)): \(\mu > 280\)

  2. Determine the test statistic:

    • Use the z-test for the population mean since the population standard deviation is known.

  3. Calculate the p-value:

    • Compare the p-value with the significance level \(\alpha = 0.08\).

  4. Make a decision:

    • If the p-value is less than \(\alpha\), reject the null hypothesis.
Step 1: State the Hypotheses

We need to test the claim that the mean score for the state's eighth graders on the exam is more than 280.

  • Null hypothesis (\(H_0\)): \(\mu = 280\)
  • Alternative hypothesis (\(H_a\)): \(\mu > 280\)
Step 2: Calculate the Test Statistic

We use the z-test for the population mean since the population standard deviation is known.

\[ z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}} \]

Given:

  • \(\bar{x} = 288\)
  • \(\mu = 280\)
  • \(\sigma = 36\)
  • \(n = 84\)

\[ z = \frac{288 - 280}{\frac{36}{\sqrt{84}}} \approx 2.0367 \]

Step 3: Calculate the p-value

The p-value for a one-tailed test is calculated as:

\[ p\text{-value} = 1 - \Phi(z) \]

Where \(\Phi(z)\) is the cumulative distribution function of the standard normal distribution.

\[ p\text{-value} \approx 1 - \Phi(2.0367) \approx 0.02084 \]

Step 4: Make a Decision

Compare the p-value with the significance level \(\alpha = 0.08\).

\[ p\text{-value} = 0.02084 < \alpha = 0.08 \]

Since the p-value is less than the significance level, 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 on the exam is more than 280.

\(\boxed{\text{Reject } H_0}\)

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