A fitness club claims that the average age of its members is 42

1. A fitness club claims that the average age of its members is 42. Assume that the ages are normally distributed. You believe that the average age is less than 42, therefore, you ask the ages of a few of the fitness club’s members and get the following values: 40, 55, 50, 33, 38, 35, 30, and 45.
— Do you have any reason to think the fitness club’s claim is incorrect? Using you knowledge on hypothesis testing, explain your conclusion carefully with detailed procedure. 
— Find the p-value of your test, and interpret the results. 
— Researchers will state the hypothesis they would like to confirm as alternative hypothesis, why?

2. Use the following example to explain this question, why do you only need to calculate a critical value for a simple hypothesis i.e. μ = c, even if your null is μ ≥ c or μ <= c?
An engineering research center claims that, through the use of a new computer control system, automobiles should achieve, on average, at least an additional 5 miles per gallon of gas. A random sample of 81 automobiles was used to evaluate this product. The sample mean increase in miles per gallon achieved was 4.5, and the sample standard deviation was 2.9 miles per gallon.
State the appropriate null and alternative hypotheses and answer the question.

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