The President of an auto dealership believes there is a relationship between the
ID: 3061185 • Letter: T
Question
The President of an auto dealership believes there is a relationship between the number of new cars sold and the number of sales people on duty. To investigate, he selects a sample of 8 weeks and observes the number of cars sold and the number of sales people on duty for that week as follows:
1.Compute a 95% interval for mean car sales for weeks when 9 sales people are on duty.
2.In checking the work schedules for next week, the President finds that there are 9 sales people scheduled to be on duty. Develop a 95% interval for the number of cars sold next week.
Week
Sales Staff
Cars Sold
1
5
53
2
5
47
3
7
48
4
4
50
5
10
58
6
12
62
7
3
45
8
11
60
Week
Sales Staff
Cars Sold
1
5
53
2
5
47
3
7
48
4
4
50
5
10
58
6
12
62
7
3
45
8
11
60
Explanation / Answer
Let xi = number of Sales Staff in the ith week and yi = number of Cars Sold in the ith week.
Back-up theory in brief
The linear regression model Y = 0 + 1X + , ………………………………………..(1)
where is the error term, which is assumed to be Normally distributed with mean 0 and variance 2.
Estimated Regression of Y on X is given by: Y = 0cap + 1capX, ………………………….(2)
where
1cap = Sxy/Sxx and 0cap = Ybar – 1cap.Xbar..……………………………………………..(3)
Mean X = Xbar = (1/n)sum of xi ………………………………………….……………….(4)
Mean Y = Ybar = (1/n)sum of yi ………………………………………….……………….(5)
Sxx = sum of (xi – Xbar)2 …………………………………………………..………………………………..(6)
Syy = sum of (yi – Ybar)2 …………………………………………………..………………………………..(7)
Sxy = sum of {(xi – Xbar)(yi – Ybar)} …………………………………………………………………….………(8)
All above sums are over i = 1, 2, …., n
n = sample size ………………………………………………………………………………(9)
Estimate of 2 is given by s2 = (Syy – 1cap2Sxx)/(n - 2)…………………………………..(10)
100(1 - )% Confidence Interval (CI) for ycap at x = x0 is (0cap + 1capx0) ± tn – 2, /2xs[(1/n) + {(x0 – Xbar)2/Sxx}]…………………………………(11)
100(1 - )% Prediction Interval (PI) for ycap at x = x0 is (0cap + 1capx0) ± {tn – 2, /2 x s[1 + (1/n) + {(x0 – Xbar)2/Sxx}]}…………………………(12)
Preparatory Work
Summary of Excel Calculations
n
8
xbar
7
ybar
52.875
Sxx
82.875
Syy
288.875
Sxy
141.125
1cap
1.70286576
0cap
40.7420814
s^2
8.09301156
Part (a)
95% interval for mean car sales for weeks when 9 sales people are on duty
= 95% confidence interval for ycap at x = 9
CIYcapLB
53.22
CIYcapUB
58.92 [vide (11) above]ANSWER
Part (b)
95% interval for the number of cars sold next week.
= 95% Prediction Interval (PI) for ycap at x = 9
=
PI LB
48.55
PIUB
63.59 ANSWER
[vide (12) above]
DONE
n
8
xbar
7
ybar
52.875
Sxx
82.875
Syy
288.875
Sxy
141.125
1cap
1.70286576
0cap
40.7420814
s^2
8.09301156
Part (a)
95% interval for mean car sales for weeks when 9 sales people are on duty
= 95% confidence interval for ycap at x = 9
CIYcapLB
53.22
CIYcapUB
58.92 [vide (11) above]ANSWER
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