A multiple regression analysis of 7,878 cases was performed with 55 explanatory variables. Suppose that [tex]SSM = 16.5[/tex] and [tex]SSE = 100.8[/tex].

(a) Find the value of the [tex]F[/tex] statistic for testing the null hypothesis that the coefficients of all the explanatory variables are zero.

(b) What are the degrees of freedom of this statistic?

(c) Find bounds on the P-value using the appropriate table.

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Note: Multivariate regression analysis involves fitting a regression model with more than one explanatory variable. It helps to assess the effectiveness of one predictor on the outcome variable by controlling for the effects of other predictors.

(a) The calculated F-statistic is obtained as (16.5 / 55) / (100.8 / (7878 – 55 – 1)). (b) The degrees of freedom are 55 for regression and 7818 for error. (c) The bounds on the p-value can be determined using an F-distribution table or statistical software.

To calculate the F-statistic and its associated degrees of freedom and p-value for the multiple regression analysis, we need the sum of squares due to regression (SSR), the sum of squares due to error (SSE), and the number of observations (n) and the number of explanatory variables (k).
Given:
SSM = 16.5
SSE = 100.8
n = 7878
k = 55

(a) Calculating the F-statistic:
The F-statistic is given by the ratio of the mean square due to regression (MSR) to the mean square due to error (MSE).
MSR = SSM / degrees of freedom for regression = SSM / k
MSE = SSE / degrees of freedom for error = SSE / (n - k - 1)
F-statistic = MSR / MSE
Substituting the given values:
MSR = 16.5 / 55
MSE = 100.8 / (7878 - 55 - 1)
F-statistic = (16.5 / 55) / (100.8 / (7878 - 55 - 1))

(b) Calculating the degrees of freedom:
Degrees of freedom for regression = k
Degrees of freedom for error = n - k - 1
Substituting the given values:
Degrees of freedom for regression = 55
Degrees of freedom for error = 7878 - 55 - 1

(c) Finding bounds on the p-value:
To find the bounds on the p-value, we need to consult an F-distribution table or use statistical software. The p-value is the probability of observing a test statistic as extreme as the calculated F-statistic.
Using a statistical software or an F-distribution table, you can find the bounds on the p-value by comparing the calculated F-statistic to the critical F-value corresponding to the given degrees of freedom. The p-value will be the probability associated with the tail(s) of the F-distribution beyond the calculated F-statistic.

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