In: Statistics and Probability
The following are data on
y = quit rate per 100 employees in manufacturing
x = unemployment rate
The data are for United States and cover the period 1990-2002.
Year | Y | X |
1990 | 1.3 | 6.2 |
1991 | 1.2 | 7.8 |
1992 | 1.4 | 5.8 |
1993 | 1.4 | 5.7 |
1994 | 1.5 | 5.0 |
1995 | 1.9 | 4.0 |
1996 | 2.6 | 3.2 |
1997 | 2.3 | 3.6 |
1998 | 2.5 | 3.3 |
1999 | 2.7 | 3.3 |
2000 | 2.1 | 5.6 |
2001 | 1.8 | 6.8 |
2002 | 2.2 | 5.6 |
(a) Estimate the regression and report the results
(b) Construct a 95% confidence interval for β.
(c) Test the hypothesis H0 : β = 0 against the alternative β=0 at the 5% significance level.
(d) Test Normality of the residuals using Jarque-Bera test.
(e) What is likely to be wrong with the assumptions of the classical normal linear model in this case? Discuss.
Answer:
Given that:
The following are data on
y = quit rate per 100 employees in manufacturing
x = unemployment rate
The data are for United States and cover the period 1990-2002.
Year | y | x |
1990 | 1.3 | 6.2 |
1991 | 1.2 | 7.8 |
1992 | 1.4 | 5.8 |
1993 | 1.4 | 5.7 |
1994 | 1.5 | 5.0 |
1995 | 1.9 | 4.0 |
1996 | 2.6 | 3.2 |
1997 | 2.3 | 3.6 |
1998 | 2.5 | 3.3 |
1999 | 2.7 | 3.3 |
2000 | 2.1 | 5.6 |
2001 | 1.8 | 6.8 |
2002 | 2.2 | 5.6 |
a) Estimate the regression and report the results
1.3 | 6.2 | 1.69 | 38.44 | 8.06 |
1.2 | 7.8 | 1.44 | 60.84 | 9.36 |
1.4 | 5.8 | 1.96 | 33.64 | 8.12 |
1.4 | 5.7 | 1.96 | 32.49 | 7.98 |
1.5 | 5.0 | 2.28 | 25.00 | 7.50 |
1.9 | 4.0 | 3.61 | 16.00 | 7.60 |
2.6 | 3.2 | 6.76 | 10.24 | 8.32 |
2.3 | 3.6 | 5.29 | 12.96 | 8.28 |
2.5 | 3.3 | 6.25 | 10.89 | 8.25 |
2.7 | 3.3 | 7.29 | 10.89 | 8.91 |
2.1 | 5.6 | 4.41 | 31.36 | 11.76 |
1.8 | 6.8 | 3.24 | 46.24 | 12.24 |
2.2 | 5.6 | 4.84 | 31.36 | 13.32 |
In these equation to find a and b
b) Construct a 95% confidence interval for β.
95% CI for
c) Test the hypothesis H0 : β = 0 against the alternative β=0 at the 5% significance level.
%
p-value is 0.001<0.05
This result is significant. So we reject the null hypothesis.
Using R Programming:
d) Test Normality of the residuals using Jarque-Bera test.
From JB test
p= 0.224>0.05
That is follow normality
e)What is likely to be wrong with the assumptions of the classical normal linear model in this case? Discuss.
If X or Y from the data to be analised by simple regression model were sample violate one or more of the regression assumption when the our result of analysis be incorrect.