1. Here is a bivariate data set.
x | y |
---|---|
83.7 | 47.7 |
73.3 | 55.5 |
68.5 | 42.3 |
59 | 43.5 |
67 | 41.8 |
61.6 | 40.8 |
64.3 | 39.2 |
99.7 | 71.7 |
60.8 | 34.7 |
69.8 | 44 |
86 | 59.4 |
47.3 | 30.8 |
37.6 | 20.8 |
58.6 | 37.4 |
57.3 | 41.1 |
Find the correlation coefficient and report it accurate to three
decimal places.
r =
2.Run a regression analysis on the following bivariate set of data with y as the response variable.
x | y |
---|---|
3.8 | 74.8 |
54.7 | 49.8 |
26.5 | 64.7 |
-7.8 | 86.8 |
22.6 | 56.6 |
60.7 | 36.1 |
39.3 | 54.3 |
-3.4 | 91.2 |
30.4 | 58.8 |
50.2 | 55.2 |
32 | 52.9 |
Verify that the correlation is significant at an α=0.05. If the
correlation is indeed significant, predict what value (on average)
for the explanatory variable will give you a value of 43.3 on the
response variable.
What is the predicted explanatory value?
x =
(Report answer accurate to one decimal place.)
1:
Following table shows the calculations:
X | Y | X^2 | Y^2 | XY | |
83.7 | 47.7 | 7005.69 | 2275.29 | 3992.49 | |
73.3 | 55.5 | 5372.89 | 3080.25 | 4068.15 | |
68.5 | 42.3 | 4692.25 | 1789.29 | 2897.55 | |
59 | 43.5 | 3481 | 1892.25 | 2566.5 | |
67 | 41.8 | 4489 | 1747.24 | 2800.6 | |
61.6 | 40.8 | 3794.56 | 1664.64 | 2513.28 | |
64.3 | 39.2 | 4134.49 | 1536.64 | 2520.56 | |
99.7 | 71.7 | 9940.09 | 5140.89 | 7148.49 | |
60.8 | 34.7 | 3696.64 | 1204.09 | 2109.76 | |
69.8 | 44 | 4872.04 | 1936 | 3071.2 | |
86 | 59.4 | 7396 | 3528.36 | 5108.4 | |
47.3 | 30.8 | 2237.29 | 948.64 | 1456.84 | |
37.6 | 20.8 | 1413.76 | 432.64 | 782.08 | |
58.6 | 37.4 | 3433.96 | 1398.76 | 2191.64 | |
57.3 | 41.1 | 3283.29 | 1689.21 | 2355.03 | |
Total | 994.5 | 650.7 | 69242.95 | 30264.19 | 45582.57 |
2:
Following is the output of regression analysis:
SUMMARY OUTPUT | ||||||
Regression Statistics | ||||||
Multiple R | 0.938477378 | |||||
R Square | 0.880739789 | |||||
Adjusted R Square | 0.867488654 | |||||
Standard Error | 5.966106502 | |||||
Observations | 11 | |||||
ANOVA | ||||||
df | SS | MS | F | Significance F | ||
Regression | 1 | 2365.791977 | 2365.791977 | 66.46524 | 1.90269E-05 | |
Residual | 9 | 320.3498411 | 35.59442679 | |||
Total | 10 | 2686.141818 | ||||
Coefficients | Standard Error | t Stat | P-value | Lower 95% | Upper 95% | |
Intercept | 80.6792263 | 2.919996322 | 27.62990682 | 5.17E-10 | 74.07373572 | 87.28471688 |
x | -0.667545273 | 0.081881059 | -8.152621419 | 1.9E-05 | -0.852773096 | -0.48231745 |
The p-value of slope is 0.000. Since p-value is less than 0.05 so we can conclude that correlation coefficient between the variables is significant.
The regression equation is
y' = 80.679 - 0.668*x
The predicted value for x = 43.3 is
y' = 80.679 - 0.668*43.3 = 51.7546
Answer: 51.8
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