To compare the amounts spent in the first three months by clients of two meal-replacement diets, a researcher randomly selects 20 clients of each diet. The mean amount spent for Diet A is $643. Assume the population standard deviation is $89. The mean amount spent for Diet B is $588. Assume the population standard deviation is $75. At α = 0.01, can the researcher support the claim that the mean amount spent in the first three months by clients of Diet A is greater than the mean amount spent in the first three months by clients of Diet B?

(a) Identify the claim and state H₀, and H_a,

(b) Find the critical value(s) and identify the rejection region(s),

(c) Find the standardized test statistic z,

(d) Decide whether to reject or fail to reject the null hypothesis,

(e) Interpret the decision in the context of the original claim. Assume the samples are random and independent, and the populations are normally distributed. If convenient, use technology.