Final answer:
The critical value at an alpha of 5% for a right-tailed z-test is approximately 1.645. This value is used to determine whether to reject the null hypothesis in favor of the alternative hypothesis that the population mean is greater than 100.
Step-by-step explanation:
To answer the question regarding the critical value at alpha = 5%, we will perform a hypothesis test known as a z-test, because the population standard deviation is known, and the sample size is large enough or the population is assumed to be normally distributed. The null hypothesis (μ = 100) states that there is no difference between the population mean and the specified value of 100, while the alternative hypothesis (μ > 100) suggests that the population mean is greater than 100. Given that the test is right-tailed, we're interested in the critical value for which 5% of the distribution lies to the right.
The critical value can be found using the standard normal distribution (z-distribution). Since the significance level (α) is 0.05 for a right-tailed test, we are looking for a z-score that corresponds to the point at which 95% of the distribution lies to the left, and the remaining 5% lies to the right. The critical value for α = 0.05 in a standard normal distribution is approximately 1.645. This means that if the calculated z-statistic from the sample data is greater than 1.645, we will reject the null hypothesis in favor of the alternative hypothesis that the mean is greater than 100. If the calculated z-statistic is less than 1.645, we fail to reject the null hypothesis.