Final answer:
The critical value for Doug's test on tennis ball catching time is likely from a t-distribution due to the sample size being less than 30. However, if we are mistakenly given standard normal (Z) distribution values, the correct critical Z-value for a two-tailed test with α = 0.01 would be option C (2.508). A clarification is needed on which distribution should be used to find the correct critical value.
Step-by-step explanation:
The critical value in a hypothesis test is the point on the scale of the test statistic beyond which we reject the null hypothesis, and it's determined by the significance level, α. Here, Doug's ability to catch tennis balls in at most 4 seconds is being tested. Since the sample size is 23 (which is less than 30), we might assume the distribution of sample means to be approximately normal due to the Central Limit Theorem if the population distribution is not known to be normal. Given α = 0.01, this is a two-tailed test because we're interested in whether Doug takes too long and also whether he's faster than claimed.
Because the sample size is less than 30 and the population standard deviation is likely unknown, we'd use the t-distribution to find the critical value. Using the degrees of freedom (α = 22) and α = 0.01 (for a two-tailed test, α/2 = 0.005 for each tail), we find the critical t-value from the t-distribution table. Unfortunately, the options you've provided (A, B, C, D) seem to come from the standard normal (Z) distribution, not the t-distribution. Doubling checking on a t-distribution table with the given α and degrees of freedom would provide the correct critical t-value. However, if we go along with the standard normal distribution values given in your options and the two-tailed nature of the test, the correct critical Z-value would be the one that corresponds to the upper tail (since the meantime is greater than 4 seconds, we'd be looking at the right side of the distribution), which is option C (2.508).
It's important to clarify whether the test statistic should be from a t-distribution or Z-distribution, as the correct answer may vary based on this. If using the t-distribution, neither of the answers provided directly aligns with typical t-distribution critical values for the given degrees of freedom and significance level.