Statistics

Question 1

Table 1 gives call waiting times of 100 customers.

a) Determine the mean and standard deviation of samples.

b) How does the Empirical Rule describe the bank customer waiting times?

c) Use the Empirical Rule to calculate estimates of tolerance intervals

containing 68.26 percent, 95.44 percent, and 99.73 percent of all waiting

times.

d) Does the estimate of a tolerance interval containing 68.26 percent of all

waiting times provide evidence that at least two-thirds of all customers will

have to wait less than eight minutes for service? Explain your answer.

Table 1. call waiting times (in Minutes) of 100 customers

1.6 6.2 3.2 5.6 7.9 6.1 7.2 6.6 5.4 6.5 4.4 1.1 3.8

7.3 5.6 4.9 2.3 4.5 7.2 10.7 4.1 5.1 5.4 8.7 6.7 2.9

7.5 6.7 3.9 0.8 4.7 8.1 9.1 7.0 3.5 4.6 2.5 3.6 4.3

7.7 5.3 6.3 6.5 8.3 2.7 2.2 4.0 4.5 4.3 6.4 6.1 3.7

5.8 1.4 4.5 3.8 8.6 6.3 0.4 8.6 7.8 1.8 5.1 4.2 6.8

10.2 2.0 5.2 3.7 5.5 5.8 9.8 2.8 8.0 8.4 4.0 3.4 2.9

11.6 9.5 6.3 5.7 9.3 10.9 4.3 1.3 4.4 2.4 7.4 4.7 3.1

4.8 5.2 9.2 1.8 3.9 5.8 9.9 7.4 5.0

Question 2

Suppose that in a survey of 1,000 residents, 721 residents believed that the

amount of violent television programming had increased over the past 5 years,

454 residents believed that the overall quality of television programming had

decreased over the past 5 years, and 362 residents believed both.

a) What proportion of the 1,000 residents believed that the amount of

violent programming had increased over the past 5 years?

b) What proportion of the 1,000 residents believed that the overall quality

of programming had decreased over the past 5 years?

c) What proportion of the 1,000 residents believed that both the amount of

violent programming had increased and the overall quality of

programming had decreased over the past 5 years?

d) What proportion of the 1,000 residents believed that either the amount

of violent programming had increased or the overall quality of

programming had decreased over the past 5 years?

e) What proportion of the residents who believed that the amount of

violent programming had increased believed that the overall quality of

programming had decreased?

f) What proportion of the residents who believed that the overall quality of

programming had decreased believed that the amount of violent

programming had increased?

g) What sort of dependence seems to exist between whether residents

believed that the amount of violent programming had increased and

whether residents believed that the overall quality of programming had

decreased? Explain your answer.

Question 3

Test results obtained from a survey are normally distributed with a mean score

of 100 and a standard deviation of 16.

a) Sketch the distribution of test results.

b) Write the equation that gives the Z score corresponding to a result.

Sketch the distribution of such Z scores.

c) Find the probability that a randomly selected result, which is

(1) Over 140.

(2) Under 88.

(3) Between 72 and 128.

(4) Within 1.5 standard deviations of the mean.

d) Suppose you take the test and receive a score of 136.

What percentage of people would receive a score higher than yours?

Question 4

A customer is considered to be very satisfied with his or her XYZ Box video

game system if the customer’s composite score on the survey instrument is at

least 42. One way to show that customers are typically very satisfied is to show

that the mean of the population of all satisfaction ratings is at least 42. Letting

this mean be µ, in this exercise we wish to investigate whether the sample of 65

satisfaction ratings provides evidence to support the claim that µ exceeds 42

(and, therefore, is at least 42).

Assume that µ equals 42. It is attempted to use the sample to contradict this

assumption in favour of the conclusion that µ exceeds 42. Recall that the mean of

the sample of 65 satisfaction ratings is �̅ = 42.95 , and assume that s, the

standard deviation of the population of all satisfaction ratings, is known to be

2.64.

a) Consider the sampling distribution of for random samples of 65 customer

satisfaction ratings.

b) Use the properties of this sampling distribution to find the probability of

observing a sample mean greater than or equal to 42.95 when we assume

that µ equals 42.

c) If µ equals 42, what percentage of all possible sample means is greater

than or equal to 42.95?