A veterinary hospital manager suspects that pet owners visit the veterinarian less than 3 times a year. To verify this suspicion, he hires a market research agency that takes a random sample of 64 clients and asks them the number of times they go to the veterinarian per year. The decision the market researcher makes is that if 2.8, then he will have sufficient evidence to say that the manager's suspicion is true. Assume that the frequency of visiting the veterinarian has a standard deviation of 1.02.   a) Write the null hypothesis and the alternative hypothesis in this case.   b) What is the rejection region used by the investigator?   c) What is the probability of type I. error if this rejection region is used?   d) If the true mean of the number of visits to the veterinarian is = 2.7, what is the   probability of type II error?   e) If the true mean of the number of visits to the veterinarian is = 2.7, what is the   power of the test?   f) If the true mean of the number of visits to the veterinarian is = 2.7, what is the sample size required to obtain a power of 0.9?

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...