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1) For the purpose of statistical study of the average family income in the United States, suppose a sample of 1,000 families is selected

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1) For the purpose of statistical study of the average family income in the United States, suppose a sample of 1,000 families is selected. It can be seen that the calculation of the sample mean in this case is easily obtained since the number of families is finite and possibly very small. Consider 16 uniformly iid random variables X₁, X₂, …., X₁₆ distributed on the open interval [0,16). Let X represent this sample. What is the mean and variance of X?

2) In studying the monthly income of a family in the United States, the population mean, that is, the average income of a family, is very difficult to compute. The reason is that the number of families in the country is huge and it is not easy to reach each and every one of them to obtain their monthly income as needed to average. However, selecting a random sample is a very easy and practical way of doing it. So, this sample statistics summarizes the entire sample into a single numerical value.

3) The lifetime of a certain type of light bulb has an exponential distribution with an average value of 200 hours. A) Calculate the first median value of the lifetime of this type of bulb and interpret this value. B) What is the 90th percentile of a bulb of the above type? A certain brand of car has an average gas mileage of 35 miles per hour and a standard deviation of 5 miles per hour. If you select a car of the above brand, what is the third quartile of the gas mileage? What does this third quartile represent?

4) Consider a particular type of car engine, which has an average gas mileage of 35 miles per gallon and a standard deviation of 5 miles per gallon. Calculate the 95th percentile of the gas mileage and interpret the result. Suppose the waiting time for a bus in a particular bus routine follows a uniform random variable between 0 and 10 minutes. Calculate the 80th percentile for the waiting time. Assume that the lifetime of a car engine follows an exponential distribution with an average of 10 years. A) Calculate the median lifetime of the car engine.B) What is the 90th percentile of the car engine and interpret the result?

5) John and Sam sat for two different examinations. Suppose John’s grade was 75 and the average and the standard deviation of his examination were 60 and 10, respectively. Similarly, assume that Sam’s grade was 85 and the average and the standard deviation of his examination were 90 and 0.5, respectively.

a. Whose grade is better?

b. Calculate z-scores for both John’s and Sam’s grades?

c. Interpret both z-scores.

d. What do you think about their grades now?

6) Let us calculate the arithmetic average of the weights of a random sample of 100 students, in pound, on the campus of a university on a weekday. Consider the data set 3, 7, 6, 8, 6, 3, 3, 4, 2. Let us find its mode. For a community at random, it is determined that the family income can be approximated by a normal distribution with mean and variance as $2,000 and $10,000, respectively. Calculate the 95th percentile of the income of a family in that community and interpret it.