x3 = −16.2 + 4.0x1 + 9.4x4 − 1.3x7
Suppose that n = 14 data points were used to construct the given regression equation and that the standard error for the coefficient of x4 is 0.963. Construct a 90% confidence interval for the coefficient of x4.
Find the lower limit and upper limit. Then, using the information from the previous question and the level of significance 5%, test the the claim that the coefficient of x4 is different from zero.
To construct a confidence interval for the coefficient of x4, you can use the formula:
Upper Limit = β4 + (critical value) * SE(β4)
Lower Limit = β4 - (critical value) * SE(β4)
Here, β4 represents the coefficient of x4, SE(β4) represents the standard error for the coefficient of x4, and the critical value is obtained from a t-distribution table for the desired confidence level.
Since n = 14 data points were used, the degrees of freedom (df) can be calculated as follows:
df = n - number of predictors = 14 - 3 = 11
To determine the critical value for a 90% confidence interval and df = 11, you need to find the t-value that corresponds to an alpha level of 0.1/2 = 0.05 (since it is a two-tailed test). Consulting a t-distribution table, you find that the critical value is approximately 2.201.
Now, substituting the given information into the formula:
Upper Limit = β4 + 2.201 * 0.963
Lower Limit = β4 - 2.201 * 0.963
To test the claim that the coefficient of x4 is different from zero, you can use the t-test. The null hypothesis (H0) states that the coefficient of x4 is equal to zero, while the alternative hypothesis (Ha) states that it is different from zero.
The test statistic (t-value) can be calculated using the formula:
t = (β4 - hypothesized value) / SE(β4)
In this case, the hypothesized value is zero. So the formula becomes:
t = β4 / SE(β4)
Substituting the given information:
t = β4 / 0.963
Finally, to conduct the t-test at a significance level of 5%, you can compare the calculated t-value (from the previous step) with the critical t-value corresponding to the degrees of freedom (11) and the desired level of significance (0.05). If the calculated t-value is outside the critical t-value bounds, then you reject the null hypothesis.