Question

Spending on credit cards decreases after the Christmas spending season​ (as measured by amount charged on a credit card in​ December). The accompanying data set contains the monthly credit card charges of a random sample of 99cardholders. Complete parts​ a) through​ e) below.

December on left and January on the right

1542.99	902.92
4301.97	7208.23
4229.48	4240.16
202.62	79.93
3298.47	4040.63
874.08	89.25
3806.31	3293.15
1934.11	2419.43
99.25	83.86
503.91	6.43
410.93	0.00
683.66	563.93
2160.65	2713.77
1123.17	187.02
2506.62	3266.62
1838.22	1522.32
9.94	1358.47
2332.42	732.94
78.51	75.03
101.31	70.21
598.23	634.71
648.87	1040.59
235.97	553.45
1266.34	1017.34
2123.35	1305.54
3.66	249.23
306.35	48.71
1902.49	872.77
558.68	485.62
2447.96	616.65
2799.33	1573.35
531.52	422.78
536.58	770.33
766.98	56.54
1958.44	1486.71
1678.16	495.43
2062.18	1065.31
397.08	510.51
5646.52	5640.14
5.51	5.51
2281.11	870.86
3820.38	1635.04
89.14	92.22
1450.51	669.75
527.35	829.29
105.86	69.26
1403.84	831.72
4234.61	2300.57
632.74	270.51
970.88	210.38
348.42	1011.16
0.00	1043.93
49.96	298.57
29.99	−29.99
471.78	1636.98
1115.95	1733.31
70.66	0.00
31.07	31.41
4.95	4.95
2523.13	1088.45
16.94	26.89
40.52	120.15
259.18	2006.79
123.05	291.22
0.00	104.07
109.71	52.99
5053.41	2839.36
3675.48	675.63
139.88	221.54
75.96	37.76
3150.92	533.41
2987.39	1932.47
651.55	692.88
9128.77	6810.41
916.81	393.47
2875.81	1308.63
797.06	796.87
34.56	0.00
44.16	1039.53
478.48	564.97
762.14	339.33
2349.94	5275.61
44.29	40.07
43.25	43.37
1339.34	653.49
1127.91	1072.21
2801.17	2336.41
52.09	91.46
1294.14	1434.02
328.19	720.64
28.31	28.58
598.68	980.71
4283.98	1576.08
568.23	0.00
479.75	162.04
1616.93	493.82
285.44	533.55
1283.98	462.53
3761.71	1477.77

a) Build a regression model to predict January spending from​December's spending.

Jan=____+____DEC (Round to four decimal places as​ needed.)

Check the conditions for this model. Select all of the true statements related to checking the conditions.

A. All of the conditions are definitely satisfied.

B. The Randomization Condition is not satisfied.

C. The Nearly Normal Condition is not satisfied.

D. The Equal Spread Condition is not satisfied.

E. The Linearity Condition is not satisfied. ​

b) How​ much, on​ average, will cardholders who charged ​$2000 in December charge in​ January?

​$ ____ ​(Round to the nearest cent as​ needed.) ​

c) Give a​ 95% confidence interval for the average January charges of cardholders who charged ​$2000 in December.

($____,$____)​(Round to the nearest cent as​ needed.) ​

d) From part​ c), give a​ 95% confidence interval for the average decrease in the charges of cardholders who charged ​$2000 in December.

($____,$____)(Round to the nearest cent as​ needed.) ​

e) What​ reservations, if​ any, would a researcher have about the confidence intervals made in parts​ c) and​ d)? Select all that apply.

A. The data are not​ independent, so the confidence intervals are not valid.

B. The data are not​ linear, so the confidence intervals are not valid.

C. The residuals show increasing​ spread, so the confidence intervals may not be valid.

D. The residuals show a curvilinear​ pattern, so the confidence intervals may not be valid.

E. A researcher would not have any reservations. The confidence intervals are valid.

Answer 1

Ʃx =	132284.26
Ʃy =	104474.62
Ʃxy =	309198402.4
Ʃx² =	419276460.8
Ʃy² =	303002692.7
Sample size, n =	99
x̅ = Ʃx/n = 132284.26/99 =	1336.204646
y̅ = Ʃy/n = 104474.62/99 =	1055.299192
SSxx = Ʃx² - (Ʃx)²/n = 419276460.7956 - (132284.26)²/99 =	242517617.9
SSyy = Ʃy² - (Ʃy)²/n = 303002692.677 - (104474.62)²/99 =	192750710.6
SSxy = Ʃxy - (Ʃx)(Ʃy)/n = 309198402.3583 - (132284.26)(104474.62)/99 =	169598929.7

a) Slope, b = SSxy/SSxx = 169598929.67667/242517617.92946 = 0.699326223

y-intercept, a = y̅ -b* x̅ = 1055.29919 - (0.69933)*1336.20465 = 120.8562438

Regression equation :

JAN = 120.8562 + (0.6993) DEC   

--

Conditions:

C. The Nearly Normal Condition is not satisfied.

D. The Equal Spread Condition is not satisfied.

--

b) Predicted value of y at x = 2000

JAN = 120.8562 + (0.6993) * 2000 = $1519.51

--

c) Predicted value of y at x = 2000

ŷ = 120.8562 + (0.6993) * 2000 = 1519.5087

Significance level, α = 0.05

Critical value, t_c = T.INV.2T(0.05, 97) = 1.9847

Sum of Square error, SSE = SSyy -SSxy²/SSxx = 192750710.61494 - (169598929.67667)²/242517617.92946 = 74145731.76

Standard error, se = √(SSE/(n-2)) = √(74145731.75588/(99-2)) = 874.29342

95% Confidence interval :

Lower limit = ŷ - tc*se*√((1/n) + ((x-x̅)²/(SSxx))) = 1330.08

Upper limit = ŷ + tc*se*√( (1/n) + ((x-x̅)²/(SSxx))) = 1708.94

--

d) 95% Prediction interval :  

Lower limit = ŷ - tc*se*√(1 + (1/n) + ((x-x̅)²/(SSxx))) = -226.03

Upper limit = ŷ + tc*se*√(1 + (1/n) + ((x-x̅)²/(SSxx))) = 3265.05

e) C. The residuals show increasing​ spread, so the confidence intervals may not be valid.