Question: You have been given the following information and data that pertains the demand for firm’s product summarized as a function of its own price and income.
Where the total number of statistical units (n) is 150 and the number of parameters (k) is 3.
From the above information, compute the following values in linear regression analysis.
- A) R-Squared (R2)
- B) Adjusted R2
- D) Degrees of freedom; between, within, and total
- E) Standard error of intercept
- F) t-statistic for the price of X
- G) Coefficient t estimate of income
- A) R-Square is referred to as the statistical measure that shows how close the data is to the fitted regression line (Fox, 2015). It is known as coefficient of determination and is computed as the ratio of error sum of squares from the regression to the total error sum of squares. That is, in order to compute the value of R-Square, the value of error sum of squares from the regression, and the total error sum of squares have to be computed. And so, from the above information given, the value of;
Therefore, R-Square (R2) becomes;
This implies that there is a strong correlation between the variables and the relationship is strongly non-linear.
- B) The adjusted R2 is computed as;
- C) The value of is computed as:
- D) The degrees of freedom (df) is equal to the sum of individual degrees of freedom for each and every sample. And because each sample has degrees of freedom that are equivalent to one less than their size of sample, with samples k, the total degrees of freedom will be attained through subtracting k from the total number of sample size. That is;
- Degrees of freedom between (SSB)
- Degrees of freedom within (SSW)
- Total = SSW + SSB = N – 1 = 147 + 2 = 149
- E) To compute the standard error of intercept, you use the relationship between t-statistic and the parameter estimate value together with its standard error. That is;
- F) Similarly, t-statistic is referred to the measure of relative strength of prediction. In essence, it is more reliable than the regression coefficient since it takes into account the error value. And so, to calculate the t-statistic for the price of X, you use the relationship between the parameter estimate value together with its standard error. That is;
- G) The same method is used to estimate the value of coefficient estimate for income. And this is;
For that case, the computed values altogether are;
For each and every type of t-test one uses, a specific procedure has to be used in order to account to all your data in the sample make it a single value, the t-value (Darlington & Hayes, 2016). When you calculate the t-values, you are comparing your sample means to the hypothesis and using both the size of sample and the data variability. When t-value is 0, this indicates that the results from the sample are equivalent to the null hypothesis. However, t-value increases as a result of increase in the difference between the sample data and the null hypothesis.
In this case, the rule of thumb states that I the absolute value of t-statistic is greater than 2, the corresponding parameter estimate is statistically indistinct from zero. This implies that having worked out the values of t-statistic for both the income and intercept, we’ve noticed they are greater than 2. And so, his indicates that the estimated coefficient of the income and intercept are statistically dissimilar from zero.
Let us analyze the regression line and see how the data fits. It’s good to note that there is only a little difference between the value of R2 and that of adjusted R2, where both are 0.85. When the value of coefficient of determination is high, it implies that the data fits well in the regression line.
The significance value for this regression is considered to be 0.00, and the F-statistic has a value of 12.05, which is a large value in this case. This means that there is absolutely no chance that the estimated regression model would fit the data, and if it does, perhaps in an accidental way (Gao & King, 2015). Consequently, the regression is regarded to be highly significant.