Question

One of the major measures of quality of services provided by any organization is the speed with which it responds to customer complaints. A large family- held department store selling furniture and flooring, including carpet, had undergone a major expansion in the past several years. In particular, the flooring department had expanded from 2 installation crews. The store had the business objective of improving its response to complaints. The variable of interests was defined as the number of days between when the complaint was made and when it was resolved. Data were collected from 50 complaints that were made in the past year. These data, stored in furniture, are:

54 5 35 137 31 27 152 2 123 81 74 27

11 19 126 110 110 29 61 35 94 31 26 5

12 4 165 32 29 28 29 26 25 1 14 13

13 10 5 27 4 52 30 22 36 26 20 23

33 68

a) The installation supervisor claims that the mean number of days between the receipt of a complaint and the resolution of the complaint is 20 days. At the 0.05 level of significance, is there evidence that the claim is not true?

b) What assumption about the population distribution is needed in order to conduct the t-test in (a)?

c) construct a boxplot or a normal probability plot to evaluate the assumption made in (b)?

d) Do you think that the assumption needed in order to conduct the t-test in (a) is valid? Explain.

Answer 1

One of the major measures of quality of services provided by any organization is the speed with which it responds to customer complaints. A large family- held department store selling furniture and flooring, including carpet, had undergone a major expansion in the past several years. In particular, the flooring department had expanded from 2 installation crews. The store had the business objective of improving its response to complaints. The variable of interests was defined as the number of days between when the complaint was made and when it was resolved. Data were collected from 50 complaints that were made in the past year. These data, stored in furniture, are:

54 5 35 137 31 27 152 2 123 81 74 27 11 19 126 110 110 29 61 35 94 31 26 5 12 4 165 32 29 28 29 26 25 1 14 13 13 10 5 27 4 52 30 22 36 26 20 23 33 68

a) The installation supervisor claims that the mean number of days between the receipt of a complaint and the resolution of the complaint is 20 days. At the 0.05 level of significance, is there evidence that the claim is not true?

single sample t test is used

Two sided test

=3.8858                

Table value of t with 48 DF at 0.05 level =2.0096

Rejection Region: Reject Ho if t < -2.0096 or t > 2.0096

Calculated t = 3.8858 in the rejection region

The null hypothesis is rejected.

There is sufficient evidence to reject the claim that the mean number of days between the receipt of a complaint and the resolution of the complaint is 20 days.

t Test for Hypothesis of the Mean
Data
Null Hypothesis                m=	20
Level of Significance	0.05
Sample Size	50
Sample Mean	43.04
Sample Standard Deviation	41.92605736
Intermediate Calculations
Standard Error of the Mean	5.9292
Degrees of Freedom	49
t Test Statistic	3.8858
Two-Tail Test
Lower Critical Value	-2.0096
Upper Critical Value	2.0096
p-Value	0.0003
Reject the null hypothesis

b) What assumption about the population distribution is needed in order to conduct the t-test in (a)?

The assumptions are:

The data are random and independent.

The data come from normal population.

c) construct a boxplot or a normal probability plot to evaluate the assumption made in (b)?

The box plot shows the data are positively skewed. This means the assumption of normality is violated.

d) Do you think that the assumption needed in order to conduct the t-test in (a) is valid? Explain.

In this case, that the normality assumption is not needed in order to conduct the t-test because the sample size 50 is large ( >30). There fore by central limit theorem, the sampling distribution sample mean is approximately normally distributed.