Final answer:
To test whether a batch of quarters is counterfeit, a one-sample z-test is used, with the null hypothesis stating the mean weight is 5.67 grams. The rejection region is determined at a 0.01 significance level, and a test statistic and p-value are computed. Based on the p-value, we can reject or fail to reject the null and make an inference about the authenticity of the coins.
Step-by-step explanation:
To determine whether the batch of quarters is counterfeit, a specific hypothesis test should be used. Given a known population standard deviation and that the sample size is less than 30, a one-sample z-test for means is appropriate.
Hypothesis for Testing
The null hypothesis (H0) is that the mean weight of the batch of quarters is 5.67 grams, the same as that of genuine quarters. The alternative hypothesis (Ha) is that the mean weight is not 5.67 grams, suggesting the quarters might be counterfeit.
Rejection Region
For a significance level (α) of 0.01 and given that the population standard deviation is 0.02, the rejection region is calculated using the standard normal distribution. The z-values that correspond to the upper and lower 0.5% of the distribution are approximately ±2.58.
Test Statistic and P-value
The test statistic is a z-score calculated from the sample mean and population standard deviation. The p-value is then obtained from the standard normal distribution to determine the likelihood of observing a test statistic as extreme as the one calculated if the null hypothesis is true.
Inference
After calculating the test statistic and the p-value, we compare the p-value to α, if p-value < (alpha = 0.01), we reject the null hypothesis, indicating that there is sufficient evidence to suggest the coins may be counterfeit. If the p-value is greater than α, we do not reject the null hypothesis.