[DATA] CEO Performance (Refer to Problem 33 in Section 4.1) The following data represent the total compensation for 12 randomly selected chief executive officers (CEOs) and the company’s stock performance in 2017.

b. Assuming the residuals are normally distributed, test whether a linear relation exists between compensation and stock return at the alpha = 0.05 level of significance

Verified step by step guidance

1. Step 1: Define the hypotheses for the linear regression test. The null hypothesis \(H_0\) states that there is no linear relationship between CEO compensation and stock return, which means the slope \(\beta_1 = 0\). The alternative hypothesis \(H_a\) states that there is a linear relationship, so \(\beta_1 \neq 0\).
2. Step 2: Calculate the least squares regression line for the data, which has the form \(\hat{y} = \hat{\beta}_0 + \hat{\beta}_1 x\), where \(x\) is the compensation and \(y\) is the stock return. This involves computing the slope \(\hat{\beta}_1\) and intercept \(\hat{\beta}_0\) using the formulas: \(\hat{\beta}_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}\) \(\hat{\beta}_0 = \bar{y} - \hat{\beta}_1 \bar{x}\)
3. Step 3: Calculate the residuals for each data point, which are the differences between observed values and predicted values: \(residual_i = y_i - \hat{y}_i\) Then, compute the residual standard error (standard deviation of residuals) to estimate the variability of the errors.
4. Step 4: Perform the hypothesis test for the slope \(\beta_1\). Calculate the test statistic using \(t = \frac{\hat{\beta}_1}{SE(\hat{\beta}_1)}\) where \(SE(\hat{\beta}_1)\) is the standard error of the slope estimate. This standard error can be found using the residual standard error and the variability in \(x\) values.
5. Step 5: Determine the critical value or p-value from the t-distribution with \(n-2\) degrees of freedom (where \(n=12\)). Compare the test statistic to the critical value or compare the p-value to the significance level \(\alpha = 0.05\). If the test statistic falls in the rejection region or the p-value is less than 0.05, reject the null hypothesis and conclude that a linear relationship exists between compensation and stock return.

Key Concepts

Linear Regression and Correlation

Linear regression models the relationship between two variables by fitting a linear equation to observed data. Correlation measures the strength and direction of this linear relationship. Understanding these helps determine if compensation and stock return move together in a predictable way.

Hypothesis Testing for Regression Slope

This involves testing whether the slope of the regression line is significantly different from zero, indicating a linear relationship. The null hypothesis states no relationship (slope = 0), while the alternative suggests a relationship (slope ≠ 0). The test uses a significance level (alpha) to decide if results are statistically meaningful.

Normality of Residuals

Residuals are the differences between observed and predicted values in regression. Assuming residuals are normally distributed is crucial for valid hypothesis testing and confidence intervals. This assumption ensures the reliability of p-values and test statistics in regression analysis.