Questions: Are you smarter than a second-grader? A random sample of 50 second-graders in a certain school district are given a standardized mathematics skills test. The sample mean score is x̄=46. Assume the standard deviation of test scores is σ=15. The nationwide average score on this test is 50. The school superintendent wants to know whether the second-graders in her school district have different math skills from the nationwide average. Use the α=0.10 level of significance and the P-value method with the π - 84 calculator. State the appropriate null and alternate hypotheses. H0: H1: This hypothesis test is a (Choose one) test.

Solution Steps

We are testing whether the mean score of second-graders in the school district is different from the nationwide average score. The hypotheses are stated as follows:

\[ \begin{align_} H_{0}: & \quad \mu = 50 \\ H_{1}: & \quad \mu \neq 50 \end{align_} \]

The standard error (\(SE\)) is calculated using the formula:

\[ SE = \frac{\sigma}{\sqrt{n}} = \frac{15}{\sqrt{50}} \approx 2.1213 \]

The Z-test statistic is calculated using the formula:

\[ Z = \frac{\bar{x} - \mu_0}{SE} = \frac{46 - 50}{2.1213} \approx -1.8856 \]

For a two-tailed test, the P-value is calculated as:

\[ P = 2 \times (1 - T(|z|)) \approx 0.0593 \]

We compare the P-value to the significance level (\(\alpha = 0.10\)):

• If \(P < \alpha\), we reject the null hypothesis.
• If \(P \geq \alpha\), we fail to reject the null hypothesis.

Since \(0.0593 < 0.10\), we reject the null hypothesis.

Final Answer