Question

Statistics Problem:

You have been asked to engage in one final project for the political organization for which you have been working. This time you wish to study the nature of the relationship between the ages of the donors to the campaign and the amount of money they plan to donate or have donated. Data is collected from a random sample of supporters of the candidate. This data is shown on the next page. Various questions need to be answered about the results you generate. These questions follow the presentation of the data. You should answer the questions posed in the narrative that should accompany your results.

Data:

                        Age of Supporter                              Donation

                                    22                                            $   75

                                    38                                               135

                                    50                                               100

                                    46                                                 50

                                    60                                               200

                                    28                                                   0

                                    25                                                 10

                                    69                                                 35

                                    75                                                 75

                                    28                                               100

                                    55                                               250

                                    37                                               100

                                    36                                               100

                                    43                                               125

                                    35                                                   0

                                    19                                                   0

                                    48                                                 50

                                    70                                                 25

                                    31                                               115

                                    30                                               105

Questions:

l)          Find a 95% confidence interval for the slope of the population regression line. Provide an explanation of the meaning of this interval. Show how this interval can be used in order to answer the question posed (at the 5% level of significance) in the previous part of the problem.

m)       Find a 95% confidence interval for the mean donation for supporters of the candidate who are 30 years of age. Provide an explanation of the meaning of this interval.

n)        Suppose a 95% confidence interval for the mean donation for supporters of the candidate who is 50 years of age is desired. Find this interval, explain its meaning, and compare its width to that of the interval computed in the previous part of the problem. Give a reason for any difference in the width between this interval and the one in the previous part of the problem.

o)        Find a 95% prediction interval for the donation of an individual supporter of the candidate who is 30 years of age. Explain its meaning relative to the problem. Compare the width of this interval to that of the interval calculated in part m) of the problem. Give a reason for any difference in the width of this interval and the one you found in part m) earlier.

Answer 1

X	Y	XY	X²	Y²
22	75	1650	484	5625
38	135	5130	1444	18225
50	100	5000	2500	10000
46	50	2300	2116	2500
60	200	12000	3600	40000
28	0	0	784	0
25	10	250	625	100
69	35	2415	4761	1225
75	75	5625	5625	5625
28	100	2800	784	10000
55	250	13750	3025	62500
37	100	3700	1369	10000
36	100	3600	1296	10000
43	125	5375	1849	15625
35	0	0	1225	0
19	0	0	361	0
48	50	2400	2304	2500
70	25	1750	4900	625
31	115	3565	961	13225
30	105	3150	900	11025

Ʃx =	845
Ʃy =	1650
Ʃxy =	74460
Ʃx² =	40913
Ʃy² =	218800
Sample size, n =	20
x̅ = Ʃx/n = 845/20 =	42.25
y̅ = Ʃy/n = 1650/20 =	82.5
SSxx = Ʃx² - (Ʃx)²/n = 40913 - (845)²/20 =	5211.75
SSyy = Ʃy² - (Ʃy)²/n = 218800 - (1650)²/20 =	82675
SSxy = Ʃxy - (Ʃx)(Ʃy)/n = 74460 - (845)(1650)/20 =	4747.5

Slope, b = SSxy/SSxx = 4747.5/5211.75 = 0.9109224

y-intercept, a = y̅ -b* x̅ = 82.5 - (0.91092)*42.25 = 44.013527

Regression equation :

ŷ = 44.0135 + (0.9109) x

Sum of Square error, SSE = SSyy -SSxy²/SSxx = 82675 - (4747.5)²/5211.75 = 78350.3957

Standard error, se = √(SSE/(n-2)) = √(78350.39574/(20-2)) = 65.97575

df = n-2 = 18

l)

Significance level, α =    0.05

Critical value, t_c = T.INV.2T(0.05, 18) = 2.1009  

95% confidence interval for the slope:

As the confidence interval contains 0. so we fail to reject the null hypothesis and conclude that there is not a significant relationship between variables.

-------

m)

Predicted value of y at x = 30

ŷ = 44.0135 + (0.9109) * 30 = 71.3412  

95% confidence interval for the mean donation for supporters of the candidate who are 30 years of age:

-----

n)

Predicted value of y at x = 50

ŷ = 44.0135 + (0.9109) * 50 = 89.5596  

95% confidence interval for the mean donation for supporters of the candidate who is 50 years of age:

The width of the interval for 30 years of age is wider than the 50 years of age confidence interval.

-----

o) 95% prediction interval for the donation of an individual supporter of the candidate who is 30 years of age:

Prediction interval is always wider than the confidence interval.