In: Statistics and Probability
Consider the data.
xi |
1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|
yi |
4 | 6 | 5 | 12 | 13 |
(a)
Compute the mean square error using equation s2 = MSE =
SSE |
n − 2 |
. (Round your answer to two decimal places.)
(b)
Compute the standard error of the estimate using equation s =MSE=
|
=
. (Round your answer to three decimal places.)
(c)
Compute the estimated standard deviation of b1 using equation sb1 =
s | ||
|
. (Round your answer to three decimal places.)
(d)
Use the t test to test the following hypotheses (α = 0.05):
H0: | β1 | = | 0 |
Ha: | β1 | ≠ | 0 |
Find the value of the test statistic. (Round your answer to three decimal places.)
test statistic=
Find the p-value. (Round your answer to four decimal places.)
p-value =
State your conclusion.
Do not reject H0. We conclude that the relationship between x and y is significant.
Reject H0. We cannot conclude that the relationship between x and y is significant.
Do not reject H0. We cannot conclude that the relationship between x and y is significant.
Reject H0. We conclude that the relationship between x and y is significant.
(e)
Use the F test to test the hypotheses in part (d) at a 0.05 level of significance. Present the results in the analysis of variance table format.
Set up the ANOVA table. (Round your values for MSE and F to two decimal places, and your p-value to three decimal places.)
Source of Variation |
Sum of Squares |
Degrees of Freedom |
Mean Square |
F | p-value |
---|---|---|---|---|---|
Regression | |||||
Error | |||||
Total |
Find the value of the test statistic. (Round your answer to two decimal places.)
Find the p-value. (Round your answer to three decimal places.)
p-value =
State your conclusion.
Do not reject H0. We cannot conclude that the relationship between x and y is significant.
Do not reject H0. We conclude that the relationship between x and y is significant.
Reject H0. We cannot conclude that the relationship between x and y is significant.
Reject H0. We conclude that the relationship between x and y is significant.
SSE =Syy-(Sxy)2/Sxx= | 12.400 |
a)
s2 =SSE/(n-2)= | 4.13 |
b)
std error σ = | =se =√s2= | 2.033 |
c)
estimated std error of slope =se(β1) =s/√Sxx= | 0.643 |
d)
test stat t = | (bo-β1)/se(β1)= | = | 3.733 |
p value: | = | 0.0168 |
Reject H0. We conclude that the relationship between x and y is significant.
Source | SS | df | MS | F | p value |
regression | 57.60 | 1 | 57.60 | 13.94 | 0.034 |
Residual error | 12.40 | 3 | 4.13 | ||
Total | 70.00 | 4 |
value of the test statistic =13.94
p value =0.034
Reject H0. We conclude that the relationship between x and y is significant.