One of the major measures of the quality of service provided

One of the major measures of the quality of service provided by any organization is the speed with which it responds to customer complaints. A large family held department store selling furniture and flooring had undergone a major expansion in the past several years. In particular, the flooring department had expanded from 2 installation crews to an installation supervisor, measurer, and 15 installation crews. The store had the business objectives of improving its response to complaints. The variable of interest 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 last year. It follows: 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 25 26 1 14 13 13 10 5 27 4 52 30 22 36 26 20 23 33 68 a. construct a 95% confidence interval estimate for the popultion mean number of days between the receipt of a complaint and the resolution of the complaint. b. what assumption must you make about the population distribution in order to construct the confidence interval estimate in (a)? c. do you think that the assumption needed in order to construct the confidence interval estimate in (a) is valid? Explain d. what effect might your conclusion in (c) have on the validity of the results in (a)?

Solution

A)

Note that              
              
Lower Bound = X - z(alpha/2) * s / sqrt(n)              
Upper Bound = X + z(alpha/2) * s / sqrt(n)              
              
where              
alpha/2 = (1 - confidence level)/2 =    0.025          
X = sample mean =    43.04          
z(alpha/2) = critical z for the confidence interval =    1.959963985          
s = sample standard deviation =    41.92605736          
n = sample size =    50          
              
Thus,              
              
Lower bound =    31.41890335          
Upper bound =    54.66109665          
              
Thus, the confidence interval is              
              
(   31.41890335   ,   54.66109665   ) [ANSWER]

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b)

We assume that this sample came from a normally distributed population.

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c)

Yes, this sample seems to be normally distributed.

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d)

The interval would have been wider had we used the t distribution when the assumption in part c is not valid.