A university claims that the average SAT math score of its incoming freshmen is 600. A skeptical education researcher believes this might not be accurate. The researcher collects a random sample of 40 students and finds a sample mean SAT math score of 622. The population standard deviation is known to be 70. Using a significance level of $\alpha$ = 0.05, test the researcher’s claim.

• A. Since P-val > $\(\alpha\)$, we fail to reject null hypothesis. There is NOT enough evidence that SAT math scores do NOT have an average of 600.
• B. Since P-val < $\(\alpha\)$, we fail to reject null hypothesis. There is NOT enough evidence that SAT math scores do NOT have an average of 600.
• C. Since P-val > $\(\alpha\)$, we reject null hypothesis. There is enough evidence that SAT math scores do NOT have an average of 600.
• D. Since P-val < $\alpha$, we reject null hypothesis. There is enough evidence that SAT math scores do NOT have an average of 600.

Answer: (D)

Verified step by step guidance

1. Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis is H₀: μ = 600, which states that the average SAT math score is 600. The alternative hypothesis is H₁: μ ≠ 600, which states that the average SAT math score is not 600.
2. Step 2: Identify the test statistic formula for a z-test since the population standard deviation is known. The formula is: $z=\frac{(\mathrm{x̄}-\mu )}{\frac{\sigma }{\sqrt{n}}}$, where x̄ is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.
3. Step 3: Substitute the given values into the formula. Here, x̄ = 622, μ = 600, σ = 70, and n = 40. Calculate the z-test statistic using the formula provided in Step 2.
4. Step 4: Determine the critical z-value for a two-tailed test at a significance level of α = 0.05. For α = 0.05, the critical z-values are approximately ±1.96. Compare the calculated z-test statistic to these critical values to decide whether to reject or fail to reject the null hypothesis.
5. Step 5: Calculate the p-value corresponding to the z-test statistic. If the p-value is less than α = 0.05, reject the null hypothesis. If the p-value is greater than α = 0.05, fail to reject the null hypothesis. Interpret the result in the context of the problem.