In: Math
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, including carpet, 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, a measurer, and 15 installation crews. The store had the business objective 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 40 complaints that were made in the past year (furniture2.xlsx). (a) The installation supervisor claims that the mean number of days between the receipt of a complaint and the resolution of the complaint is 30 days. To test the claim, build null and alternative hypotheses (for a two-tail test). (b) To conduct a two-tail t test based on the hypotheses in (a), identify the rejection regions (two sides) given the 99% critical level. (c) Using the given data, compute the test statistic for the t test. (d) At the 0.01 level of significance, should the claim be rejected (i.e., the mean number of days is different from 30)? In the critical-value approach, what is your conclusion based on (b) and (c)? Explain. (e) Using the test statistic in (c), determine the p-value for the t test. (f) In the p-value approach, what is your conclusion based on (e)? Explain.
Days |
65 |
43 |
35 |
137 |
31 |
27 |
152 |
22 |
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 |
(a)
The installation supervisor claims that the Mean number of days between the receipt of a complaint and the resolution of the complaint is 30 days.
Hypothesied Mean : = 30 days
Null hypothesis : Ho :
Alternate Hypothesis : Ha : (Two tailed test)
(b) Two tailed test critical values :
Degrees of freedom = n-1 = 40-1=39
Critical regions , at
c)
Data were collected from 40 complaints that were made in the past year;
Sample size : n= 40
Sample Data:
x : Days : number of days between the receipt of a complaint and the resolution of the complaint
x: Days | |||
65 | 17.325 | 300.1556 | |
43 | -4.675 | 21.85563 | |
35 | -12.675 | 160.6556 | |
137 | 89.325 | 7978.956 | |
31 | -16.675 | 278.0556 | |
27 | -20.675 | 427.4556 | |
152 | 104.325 | 10883.71 | |
22 | -25.675 | 659.2056 | |
123 | 75.325 | 5673.856 | |
81 | 33.325 | 1110.556 | |
74 | 26.325 | 693.0056 | |
27 | -20.675 | 427.4556 | |
11 | -36.675 | 1345.056 | |
19 | -28.675 | 822.2556 | |
126 | 78.325 | 6134.806 | |
110 | 62.325 | 3884.406 | |
110 | 62.325 | 3884.406 | |
29 | -18.675 | 348.7556 | |
61 | 13.325 | 177.5556 | |
35 | -12.675 | 160.6556 | |
94 | 46.325 | 2146.006 | |
31 | -16.675 | 278.0556 | |
26 | -21.675 | 469.8056 | |
5 | -42.675 | 1821.156 | |
12 | -35.675 | 1272.706 | |
4 | -43.675 | 1907.506 | |
165 | 117.325 | 13765.16 | |
32 | -15.675 | 245.7056 | |
29 | -18.675 | 348.7556 | |
28 | -19.675 | 387.1056 | |
29 | -18.675 | 348.7556 | |
26 | -21.675 | 469.8056 | |
25 | -22.675 | 514.1556 | |
1 | -46.675 | 2178.556 | |
14 | -33.675 | 1134.006 | |
13 | -34.675 | 1202.356 | |
13 | -34.675 | 1202.356 | |
10 | -37.675 | 1419.406 | |
5 | -42.675 | 1821.156 | |
27 | -20.675 | 427.4556 | |
Total | 1907 | 78732.78 | |
Sample mean | =1907/40=47.675 |
Hypothesied Mean : | 30 |
Sample Mean : | 47.675 |
Sample Standard Deviation : s | 44.931 |
Sample Size : n | 40 |
Level of significance : | 0.01 |
Degrees of Freedom : n-1 | 39 |
d). Critical regions , at
As Calculated Value is with in the Critical Values i.e.( -2.7079 < 2.488 < 2.7079 )Fail To Reject Null Hypothesis
Claim : that the Mean number of days between the receipt of a complaint and the resolution of the complaint is 30 days
Should the claim( that the Mean number of days between the receipt of a complaint and the resolution of the complaint is 30 days) be rejected : noe)
tstat = 2.488 (from c)
p-value for the t-test
As P-Value i.e. is greater than Level of significance i.e (P-value:0.0172 > 0.01:Level of significance); Fail to Reject Null Hypothesis
f)
As P-Value i.e. is greater than Level of significance i.e (P-value:0.0172 > 0.01:Level of significance); Fail to Reject Null Hypothesis
Should the claim( that the Mean number of days between the receipt of a complaint and the resolution of the complaint is 30 days) be rejected : no
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p-value is ccomputed using excel function
"T.DIST.RT function
Returns the right-tailed Student's t-distribution.
The t-distribution is used in the hypothesis testing of small sample data sets. Use this function in place of a table of critical values for the t-distribution.
Syntax
T.DIST.RT(x,deg_freedom)
The T.DIST.RT function syntax has the following arguments:
• X Required. The numeric value at which to evaluate the distribution.
• Deg_freedom Required. An integer indicating the number of degrees of freedom.