Problem 20 - Basic Statistics for Business and Economics with Student

Question

The Times-Observer is a daily newspaper in Metro City. Like many city newspapers, the Times-Observer is suffering through difficult financial times. The circulation manager is studying other papers in similar cities in the United States and Canada. She is particularly interested in what variables relate to the number of subscriptions to the paper. She is able to obtain the following sample information on 25 newspapers in similar cities. The following notation is used: Sub \(=\) Number of subscriptions (in thousands). Popul = The metropolitan population (in thousands). Adv \(=\) The advertising budget of the paper (in \(\$$ hundreds). Income \)=$ The median family income in the metropolitan area (in \$ thousands). a. Determine the regression equation. b. Conduct a global test of hypothesis to determine whether any of the regression coefficients are not equal to zero. c. Conduct a test for the individual coefficients. Would you consider deleting any coefficients? d. Determine the residuals and plot them against the fitted values. Do you see any problems? e. Develop a histogram of the residuals. Do you see any problems with the normality assumption?

Step-by-Step Solution

Answer

Compute the regression equation, perform global and individual hypothesis tests, analyze residuals for randomness and normality.

Step 1: Gather and Organize Data

We begin by listing the variables given: Sub (Number of subscriptions), Popul (Metropolitan population), Adv (Advertising budget), and Income (Median family income). These represent the independent and dependent variables for our regression analysis.

Step 2: Compute the Regression Equation

To determine the regression equation, use the formula for multiple linear regression, which is \[ Sub = \beta_0 + \beta_1 \cdot Popul + \beta_2 \cdot Adv + \beta_3 \cdot Income \]Calculate the coefficients \(\beta_0\), \(\beta_1\), \(\beta_2\), and \(\beta_3\) using statistical software or by solving normal equations, typically involving matrix algebra.

Step 3: Global Hypothesis Test

Perform a global F-test to test the hypothesis:\[ H_0: \beta_1 = \beta_2 = \beta_3 = 0 \]\[ H_a: \text{At least one } \beta_i eq 0 \]Calculate the F-statistic and compare it to the critical value from the F-distribution to determine if any of the regression coefficients are significantly different from zero.

Step 4: Test Individual Coefficients

Conduct t-tests for each individual regression coefficient. The hypotheses for each test are:\[ H_0: \beta_i = 0 \]\[ H_a: \beta_i eq 0 \]If any \( p \)-value is greater than a significance level (say 0.05), consider whether that variable should be removed from the model.

Step 5: Calculate Residuals and Plot Against Fitted Values

Compute the residuals for each observation, which are the differences between the observed and predicted subscriptions. Plot these residuals against the fitted values. Look for patterns such as non-random distribution, which may indicate a problem with the model fit.

Step 6: Create a Histogram of Residuals

Generate a histogram of the residuals to assess their distribution. Check for normality by seeing if the histogram resembles a bell curve. Deviations from normality could signal issues with the assumption of normally distributed errors.

Key Concepts

Hypothesis Testing

Hypothesis testing is a fundamental method in statistics that allows us to make decisions about a population parameter based on sample data. In the context of multiple regression analysis, hypothesis testing helps us determine if relationships between the independent variables and the dependent variable are statistically significant.

In a regression model, we commonly perform two types of tests: the global F-test and the individual t-tests.

The **global F-test** examines whether at least one predictor variable is associated with the outcome variable. It tests the null hypothesis: \( H_0: \beta_1 = \beta_2 = \beta_3 = 0 \) against the alternative hypothesis: \( H_a: \text{At least one } \beta_i eq 0 \).

If the F-statistic is significantly larger than the critical value from the F-distribution at a chosen significance level, we reject the null hypothesis, indicating that at least one of the coefficients is different from zero. This means that the model has at least one predictor that is significantly impacting the response variable.

On the other hand, **individual t-tests** aim to evaluate each predictor variable's contribution to the model by testing hypotheses such as: \( H_0: \beta_i = 0 \) vs. \( H_a: \beta_i eq 0 \).

Here, a significant t-test implies that the corresponding variable meaningfully contributes to predicting the outcome.

Residual Analysis

Residual analysis is an essential part of evaluating the fit of a regression model. Residuals are the differences between the observed values and the values predicted by the regression model. These are crucial because they allow us to check the consistency and validity of the model assumptions.

When **plotting residuals against fitted values**, it's important to analyze the pattern. Ideally, residuals should appear randomly scattered without forming any discernible pattern. A clear pattern could indicate non-linearity, incorrect model form, or presence of outliers.

A **random dispersion** of residuals generally suggests that the model fits well and the assumptions are met.

Moreover, a histogram of the residuals can be created to **test the normality assumption**. A normal distribution of residuals will look like a bell-shaped curve, indicating that the model's errors are normally distributed.

Deviation from this shape could suggest potential issues, such as skewness or kurtosis, meaning the model may need to be reconsidered or that data transformations might be necessary.

F-test

The F-test is integral to regression analysis for evaluating the overall significance of a regression model. It tests if the collective effect of all independent variables on the dependent variable is significant. The test checks if at least one of the regression coefficients is not equal to zero, meaning that at least one variable significantly affects the dependent variable.

In an F-test, you calculate the F-statistic, which measures the model's explained variance against its unexplained variance. To perform this test, the following steps are performed:

• Determine the total sum of squares (SST), regression sum of squares (SSR), and residual sum of squares (SSE).
• Calculate the mean square for the regression (MSR) and the mean square for the error (MSE).
• Compute the F-statistic: \( F = \frac{MSR}{MSE} \)

You then compare the F-statistic to a critical value from the F-distribution table, based on the degrees of freedom in the model.

If the computed F-value exceeds this critical value, the null hypothesis is rejected. This result suggests that at least one predictor variable has a statistically significant impact on the dependent variable.

t-test

A t-test in the context of multiple regression is used to determine the statistical significance of individual regression coefficients. Each coefficient in the regression equation is tested to see if it significantly differs from zero. A significant result indicates that the corresponding variable is a meaningful predictor of the dependent variable.

For each coefficient, the hypotheses are:\( H_0: \beta_i = 0 \)and\( H_a: \beta_i eq 0 \).

The test statistic is calculated as follows: \[ t = \frac{b_i}{\text{SE}(b_i)} \]where:

• \( b_i \) is the estimated regression coefficient.
• \( \text{SE}(b_i) \) is the standard error of the coefficient.

The computed t-value is compared against a critical value from the t-distribution table, corresponding to the level of significance selected (commonly 0.05) and degrees of freedom.

If the absolute t-value is greater than the critical value, the null hypothesis is rejected, revealing that the variable contributes significantly to predicting the dependent variable. This decision helps refine the model by retaining only the most impactful variables.