Sampling Distribution Problems

Problem 1

If many samples of size 15 (that is, each sample consists of 15 items) were taken from a large normal population with a mean of 18 and variance of 5, what would be the mean, variance, standard deviation and shape of the distribution of sample means? Give reasons for your answers.

Note: Variance is the square of the standard deviation.

 

Problem 2

If many samples of size 100 (that is, each sample consists of 100 items) were taken from a large non-normal population with a mean of 10 and variance of 16, what would be the mean, variance, standard deviation and shape of the distribution of sample means? Give reasons for your answers.

Note: Variance is the square of the standard deviation. 

Problem 3

Time lost due to employee absenteeism is an important problem for many companies. The human resources department of Western Electronics has studied the distribution of time lost due to absenteeism by individual employees. During a one-year period, the department found a mean of 21 days and a standard deviation of 10 days based on data for all the employees.

a)     

If you pick an employee at random, what is the probability that the number of absences for this one employee would exceed 25 days?

b)     

If many samples of 36 employees each are taken and sample means computed, a distribution of sample means would result. What would be the mean, standard deviation and shape of the distribution of sample means for samples of size 36? Give reasons for your answers.

c)      

A group of 36 employees is selected at random to participate in a program that allows a flexible work schedule, which the human resources department hopes will decrease the employee absenteeism in the future. What is the probability that the mean for the sample of 36 employees randomly selected for the study would exceed 25 days?

 

Problem 4

The amount of time a bank teller spends with each customer has a population mean
mx   = 3.1 minutes and population standard deviation
sx = 0.4 minute.

a)      What is the probability that for a randomly selected customer the service time would exceed 3 minutes?

b)      If many samples of 64 were selected, what are mean and standard error of the mean (standard deviation of sample means) expected to be? What is expected to be the shape of the distribution of sample means? Give reasons for all your answers.

c)       If a random sample of 64 customers is selected, what is the probability that the sample mean would exceed 3 minutes?