Question

During the holiday season, shoppers were asked to estimate how much money they spent on gifts for themselves. Raw data is given below. Are the reported amounts significantly less than the actual amounts as determined from the receipts?

1) Write Ho (null) and H1 (alternative); indicate which is being tested.

2) Perform the statistical test ad write answer to the original question as a statement related to the original query

2) Construct a 99% confidence interval estimate of the mean difference between reported amounts and actual amounts . Interpret the resulting confidence interval, does it contain 0?

Actual	Reported
53	26
67	45
72	54
72	49
62	35
70	41
73	41
68	49
64	38
58	31
73	44
37	19
63	32
67	37
52	29
59	33
64	39
36	19
59	30
72	48
57	32
61	33
54	28
40	23
63	42
43	23
66	34
60	31
60	34
61	34
40	26
64	48
65	48
49	29
47	29
59	35
72	44
65	39
63	40
70	50
48	31
50	38
76	55
46	27
61	44
63	44
48	26
41	26
53	30
52	28
46	23
43	24
75	54
57	32

Answer 1

The calculations are given after the Hypothesis Testing

From the data, Mean of the differences() = -22.889 and Std Deviation of differences (sd) = 4.89

The degrees of freedom (df) = n - 1 = 54 - 1 = 53

= 0.05 (Default)

The Hypothesis:

H0: = 0

H0: < 0 (Claim)

The Test Statistic:

P value: The p value (Left tailed) for t = -34.40, df = 53; p value = 0.000

The Critical Value: The critical value(Right Tailed) at = 0.05, df = 6; critical value = -1.6741

The Decision Rule: If t observed is < -t critical, then Reject H0.

Also, if p value is < , Then Reject H0.

The Decision: Since t observed (-34.40) is < -t critical (-1.6741), we reject H0.

Also, since p value (0.000) is < (0.05), we reject H0.

The Conclusion: Reject H0. There is sufficient evidence at the 95% level of significance to conclude that reported amounts are significantly lesser than the actual amounts as determined from the receipts.

_________________________________________________________________________

The 99% CI for the mean difference

The CI for the mean difference is given by ME

ME = tcritical * Sd/Sqrt(n) = 2.618 * 4.89/Sqrt(54) = 1.778

The Lower Limit = -22.889 - 1.782 = -24.667

The Upper Limit = -22.889 + 1.782 = -21.111

The 99% CI is (-24.667, -21.111)

We are 99% confident that the population mean difference of spending lies between the confidence limits -24.667 to -21.111.

No, The confidence interval does not contain 0. Therefore we will reject the null hypothesis that = 0.

__________________________________________________________________________________

Calculation for the mean and standard deviation:

Mean = Sum of observation / Total Observations

Standard deviation = SQRT(Variance)

Variance = Sum Of Squares (SS) / n - 1, where

SS = SUM(X - Mean)².

	Actual	Reported	Difference	Mean	(x - Mean)2
1	53	26	-27	-22.889	16.900321
2	67	45	-22	-22.889	0.790321
3	72	54	-18	-22.889	23.902321
4	72	49	-23	-22.889	0.012321
5	62	35	-27	-22.889	16.900321
6	70	41	-29	-22.889	37.344321
7	73	41	-32	-22.889	83.010321
8	68	49	-19	-22.889	15.124321
9	64	38	-26	-22.889	9.678321
10	58	31	-27	-22.889	16.900321
11	73	44	-29	-22.889	37.344321
12	37	19	-18	-22.889	23.902321
13	63	32	-31	-22.889	65.788321
14	67	37	-30	-22.889	50.566321
15	52	29	-23	-22.889	0.012321
16	59	33	-26	-22.889	9.678321
17	64	39	-25	-22.889	4.456321
18	36	19	-17	-22.889	34.680321
19	59	30	-29	-22.889	37.344321
20	72	48	-24	-22.889	1.234321
21	57	32	-25	-22.889	4.456321
22	61	33	-28	-22.889	26.122321
23	54	28	-26	-22.889	9.678321
24	40	23	-17	-22.889	34.680321
25	63	42	-21	-22.889	3.568321
26	43	23	-20	-22.889	8.346321
27	66	34	-32	-22.889	83.010321
28	60	31	-29	-22.889	37.344321
29	60	34	-26	-22.889	9.678321
30	61	34	-27	-22.889	16.900321
31	40	26	-14	-22.889	79.014321
32	64	48	-16	-22.889	47.458321
33	65	48	-17	-22.889	34.680321
34	49	29	-20	-22.889	8.346321
35	47	29	-18	-22.889	23.902321
36	59	35	-24	-22.889	1.234321
37	72	44	-28	-22.889	26.122321
38	65	39	-26	-22.889	9.678321
39	63	40	-23	-22.889	0.012321
40	70	50	-20	-22.889	8.346321
41	48	31	-17	-22.889	34.680321
42	50	38	-12	-22.889	118.570321
43	76	55	-21	-22.889	3.568321
44	46	27	-19	-22.889	15.124321
45	61	44	-17	-22.889	34.680321
46	63	44	-19	-22.889	15.124321
47	48	26	-22	-22.889	0.790321
48	41	26	-15	-22.889	62.236321
49	53	30	-23	-22.889	0.012321
50	52	28	-24	-22.889	1.234321
51	46	23	-23	-22.889	0.012321
52	43	24	-19	-22.889	15.124321
53	75	54	-21	-22.889	3.568321
54	57	32	-25	-22.889	4.456321
Total			-1236		1267.33333

n	54
Sum	-1236
Average	-22.889
SS	1267.333334
Variance	23.9119497
Std Dev	4.8900