Question

Refer to the data below. Suppose X is the independent variable and Y is the dependent variable. Calculate the variance of X, variance of Y, standard deviation of X, standard deviation of Y, the covariance between X and Y, the correlation coefficient between X and Y, the slope of the regression line, the Y intercept of the regression, ESS, RSS, TSS, and R-square of the regression line. Predict the value of Y when X=25. Show your work by constructing the 12-colume table.

X |
Y |

2 |
58 |

6 |
105 |

8 |
88 |

8 |
118 |

12 |
117 |

16 |
137 |

20 |
157 |

20 |
169 |

22 |
149 |

26 |
202 |

Answer #1

X | Y | (x-x mean) | (y-y mean) | x-mean)^2 | (y-mean)^2 | (X-x mean)*(y- y mean) | Y-cap = 60+5x | ycap-y-mean | (y cap-y mean)^2 | y-y cap | (y-y cap)^2 | (y-y mean)^2 | |

2 | 58 | -12 | -72 | 144 | 5184 | 864 | 70 | -60 | 3600 | -12 | 144 | 5184 | |

6 | 105 | -8 | -25 | 64 | 625 | 200 | 90 | -40 | 1600 | 15 | 225 | 625 | |

8 | 88 | -6 | -42 | 36 | 1764 | 252 | 100 | -30 | 900 | -12 | 144 | 1764 | |

8 | 118 | -6 | -12 | 36 | 144 | 72 | 100 | -30 | 900 | 18 | 324 | 144 | |

12 | 117 | -2 | -13 | 4 | 169 | 26 | 120 | -10 | 100 | -3 | 9 | 169 | |

16 | 137 | 2 | 7 | 4 | 49 | 14 | 140 | 10 | 100 | -3 | 9 | 49 | |

20 | 157 | 6 | 27 | 36 | 729 | 162 | 160 | 30 | 900 | -3 | 9 | 729 | |

20 | 169 | 6 | 39 | 36 | 1521 | 234 | 160 | 30 | 900 | 9 | 81 | 1521 | |

22 | 149 | 8 | 19 | 64 | 361 | 152 | 170 | 40 | 1600 | -21 | 441 | 361 | |

26 | 202 | 12 | 72 | 144 | 5184 | 864 | 190 | 60 | 3600 | 12 | 144 | 5184 | |

mean= | 14 | 130 | 0 | 0 | 56.8 | 1573 | 284 | 130 | 0 | 1420 | |||

sum= | 140 | 1300 | 0 | 0 | 568 | 15730 | 2840 | 1300 | 0 | 14200 | 0 | 1530 | 15730 |

variance of X = (x-mean)^2 / (n-1) = 568/(10-1)=63.11

variance of Y=(y-mean)^2 / (n-1)=15730/9=1747.778

std dev of X = sqrt [ variance of X] = sqrt 63.11 =7.944

std dev of Y=sqrt [varince of Y] = sqrt 1747.778 = 41.806

Covariance(X,Y) = SUM(xi - xmean)*(yi - ymean)/(samplesize -1)=2840/9=315.5556

correlation coefficient,r= cov (X,Y) / (std dev of X*std dev of Y) = 315.5556/(7.944*41.806) = 0.950

slope = r*std dev of Y / std dev of X = 0.95*41.806/7.944=5

y-intercept = y mean - slope* x-mean = 130-5*14=60

ESS=sum (y cap-y mean)^{2}=14200

RSS=sum(y - y cap)^{2}=1530

TSS=ESS+RSS=14200+1530=15730

R-square = r^2 = 0.950^2 = 0.902734

or

R-square = ESS/TSS = 14200/15730=0.902734

**when X=25,**

predicted value of y =60+5*X = 60+5*25 = 185

betas
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Upper 95%
Intercept
60
9.22603481
6.503336
0.000187444
38.72472558
81.27527
X Variable 1
5
0.580265238
8.616749
2.54887E-05
3.661905962
6.338094
please show me the formula on exel how the teacher got thesee
results?
Here are the following data
Restaurant
Population as x
Sales as y
1
2
58
2
6
105
3
8
88
4
8
118
5
12
117
6
16
137
7
20
157
8
20
169
9
22
149
10...

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X
Y
X-Xbar
Y-Ybar
(X-Xbar)^2
(Y-Ybar)^2
(X-Xbar)*(Y-Ybar)
3
1
8
5
10
13
15
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19
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X Y
3
20
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7
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15
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X
11
14
10
15
14
10
13
12
y
110
130
100
140
150
120
110
130
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ANOVA
df
SS
Regression
1
3348.312
Residual
8
9529.811
Total
9
12878.123
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Standard Error
t Stat
P-value
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83.280
1.689
0.030
X
148.62
38.312
1.283
0.075
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