Assets Sales 2708 1852.8 13250 9088.4 13593 4580.2 3595 91 6382 2441.3 1061 1370
ID: 3268892 • Letter: A
Question
Assets Sales 2708 1852.8 13250 9088.4 13593 4580.2 3595 91 6382 2441.3 1061 1370.1 1099 1029.4 1599 414 2797 342.7 19785 10620.1 252 966 1097 450.4 5380 338.5 1182 425.5 1663 675.4 44759 4652.1 5616 2051.8 5866 1633.7 5807 818.7 5066 2551.4 856 1429.4 4437 8826.9 6973 7036.8 840 1613.5 425 142.7 422 1127.7 812 1145.9 4816 430.9 2600 272.2 5287 501.8 3507 1632.2 1812 2604.5 26426 28271.4 587 2234.6 1580 6557.7 A business analyst is looking at a company's assets and sales to determine the relationship (if any) between the two measures. She has data (in $ million) from a random sample of 35 Fortune 500 companies, and obtained a linear regression. Use the data to find a 95% confidence interval for the slope of the regression line and interpret your interval in context. Create a 95% confidence interval for the slope of the true line. Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. (Round to two decimal places as needed.) A 95% confidence interval for beta_1 is (, ). Therefore, we can say with 95% confidence that a company's assets decrease, on average, between the upper and lower limits of the confidence interval per one million dollars in sales. A 95% confidence interval for beta_1 is (, ). Therefore, we can say with 95% confidence that a company's assets increase, on average, between the upper and lower limits of the confidence interval per one million dollars in sales. A confidence interval is not appropriate because the data do not satisfy all conditions.Explanation / Answer
Solution
Back-up Theory
Let X and Y be two variables such that Y depends on X by the model Y = + X + , where is the error term, which is assumed to be Normally distributed with mean 0 and variance 2. Then,
Y is termed ‘dependent variable’ (Assets) and
X is termed ‘independent variable’.(Sales)
Let (xi, yi) be a pair of sample observation on (X, Y), i= 1, 2, …., n, where n = sample size.
Then, Mean X = Xbar = (1/n)sum of xi over I = 1, 2, …., n; ……………….(1)
Sxx = sum of (xi – Xbar)2 over i = 1, 2, …., n ………………………………..(2)
Similarly, Mean Y = Ybar =(1/n)sum of yi over i= 1, 2, …., n;…………….(3)
Syy = sum of (yi – Ybar)2 over i = 1, 2, …., n ………………………………(4)
Sxy = sum of {(xi – Xbar)(yi – Ybar)} over i = 1, 2, …., n…………….……(5)
Estimated Regression of Y on X is given by: Y = a + bX, where
b = Sxy/Sxx and a = Ybar – b.Xbar..…………………………….………….(6)
Estimate of 2 is given by s2 = (Syy – b2Sxx)/(n - 2)
Standard Error of b is sb, where sb2 = s2/Sxx ………………………………..(7)
100(1 - )% Confidence Interval (CI) for = b ± SE(b)xtn – 2,/2 …………….(8)
Now, to work out the solution,
As shown in the Excel Calculations summary given below,
95% Confidence Interval for slope () = (0.488, 1.474) ANSWER
Since the CI has a positive boundaries, assets increase as sales increase. Hence, of the given 3 options, option (2) is the correct answer
Excel Calculations summary is given below:
n
35
xbar
3,149
ybar
5826.77143
Sxx
911380363
Syy
2645691088
Sxy
893817094
b
0.98072894
a
2738.26827
s^2
53609054.5
sb^2
0.05882182
s
7321.82044
sb
0.24253211
0.05
n-2
33
tn-2,/2
2.03451529
CIbLB
0.48729366
CIbUB
1.47416422
n
35
xbar
3,149
ybar
5826.77143
Sxx
911380363
Syy
2645691088
Sxy
893817094
b
0.98072894
a
2738.26827
s^2
53609054.5
sb^2
0.05882182
s
7321.82044
sb
0.24253211
0.05
n-2
33
tn-2,/2
2.03451529
CIbLB
0.48729366
CIbUB
1.47416422