Question

A random sample of 76 eighth grade​ students' scores on a national mathematics assessment test has a mean score of 293. 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 285. Assume that the population standard deviation is 32. At α = 0.07​, 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.
b) Find the standardized test statistic​ z, and its corresponding area.
c) Find the​ P-value.
d) Decide whether to reject or fail to reject the null hypothesis.
e) Interpret your decision in the context of the original claim.

Answer 1

The question involves testing the state school administrator's claim that the average eighth-grade math score is above 285 using hypothesis testing. This includes calculating the z-statistic and p-value to determine if there is significant evidence to support the claim based on an α of 0.07.

Step-by-step explanation:

The student's question pertains to conducting a hypothesis test to evaluate a state school administrator's claim regarding the average score of eighth graders on a mathematics assessment test. To address this, we need to define the null hypothesis (H₀) and alternative hypothesis (Hᵃ), calculate the standardized test statistic z, determine the corresponding p-value, and make a decision to either reject or fail to reject H₀ based on the significance level (α).

1. Claim mathematically: The administrator claims that the mean score (μ) is more than 285. So, H₀: μ = 285 and Hᵃ: μ > 285.
2. Standardized test statistic (z): Calculate z using the sample mean, population mean under H₀, population standard deviation, and sample size: z = (293 - 285) / (32 / √76). Calculate z and find its corresponding area.
3. p-value: Using the obtained z-value, determine the p-value, which is the probability of observing a test statistic as extreme as the z-value under the null hypothesis.
4. Decision: Reject H₀ if p-value < α; otherwise, fail to reject H₀.
5. Interpretation: If H₀ is rejected, there is enough evidence to support the claim that the mean score is more than 285. If H₀ is not rejected, there's insufficient evidence to support the claim.