A random sample of 82 eighth-grade students' scores on the national mathematics assessment test has a mean score of 282. This test result prompts state school administrators 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 40. At a = 0.11, is there enough evidence to support the administrator's claim? Complete parts (a) through (d):
(a) State the null and alternative hypotheses.
Null hypothesis: The mean score for the state's eighth graders on the exam is less than or equal to 280.
Alternative hypothesis: The mean score for the state's eighth graders on the exam is greater than 280.
(b) Compute the test statistic (z).
Test statistic (z) = (sample mean - population mean) / (population standard deviation / √(sample size))
Test statistic (z) = (282 - 280) / (40 / √82)
Test statistic (z) = 2 / (40 / 9.055)
Test statistic (z) = 2 / 4.455
Test statistic (z) = 0.449
(c) Determine the p-value associated with the test statistic.
The p-value associated with the test statistic is the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true. To determine the p-value, we need to find the area under the standard normal curve to the right of the test statistic (z = 0.449).
(d) Decide whether to reject or fail to reject the null hypothesis.
Fail to reject Ho
(e) Interpret your decision in the context of the original claim.
At the 11% significance level, there is not 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.