91Ó°ÊÓ

The following table gives information on GPAs and starting salaries (rounded to the nearest thousand dollars) of seven recent college graduates. $$ \begin{array}{l|rrrrrrr} \hline \text { GPA } & 2.90 & 3.81 & 3.20 & 2.42 & 3.94 & 2.05 & 2.25 \\ \hline \text { Starting salary } & 48 & 53 & 50 & 37 & 65 & 32 & 37 \\ \hline \end{array} $$ a. With GPA as an independent variable and starting salary as a dependent variable, compute \(\mathrm{SS}_{x x}, \mathrm{SS}_{y y}\), and \(\mathrm{SS}_{x y}\) b. Find the least squares regression line. c. Interpret the meaning of the values of \(a\) and \(b\) calculated in part b. d. Calculate \(r\) and \(r^{2}\) and briefly explain what they mean. e. Compute the standard deviation of errors. fonstruct a \(95 \%\) confidence interval for \(B\). g. Test at a \(1 \%\) significance level whether \(B\) is different from zero. h. Test at a \(1 \%\) significance level whether \(\rho\) is positive.

Step by step solution

01

Calculation

First, calculate the means of GPA (\(x\)) and the starting salary (\(y\)). Then, compute \(\mathrm{SS}_{x x}\), \(\mathrm{SS}_{y y}\), and \(\mathrm{SS}_{x y}\), using the formulas: \(\mathrm{SS}_{x x}=\sum(x_{i}-\bar{x})^{2}\), \(\mathrm{SS}_{y y}=\sum(y_{i}-\bar{y})^{2}\), \(\mathrm{SS}_{x y}=\sum(x_{i}-\bar{x})(y_{i}-\bar{y})\).

02

Regression Line

Next, find the least squares regression line using the formulas: \(b=\frac{\mathrm{SS}_{x y}}{\mathrm{SS}_{x x}}\), \(a=\bar{y}-b \bar{x}\).

03

Interpretation

The value of \(a\) represents the starting salary when GPA=0, and \(b\) represents the amount the starting salary increases for each additional unit increase in GPA.

04

Correlation Coefficient

Then, calculate \(r\) and \(r^{2}\) using the formulas: \(r=\frac{\mathrm{SS}_{x y}}{\sqrt{\mathrm{SS}_{x x} \mathrm{SS}_{y y}}}\), \((r^{2} = \frac{\mathrm{SS}^{2}_{xy}}{\mathrm{SS}_{x x}\mathrm{SS}_{y y}})\). The value of \(r\) represents the strength and direction of a linear relationship between two variables, whereas \(r^{2}\) reflects how closely the data points cluster around the regression line.

05

Standard Deviation

Next, calculate the standard deviation of errors as: \(S_{e}=\sqrt{\frac{\Sigma ( y - \widehat{y} )^{2}}{n-2}}\), where \(y\) is the actual data, \(\widehat{y}\) is the predicted data from regression line, and \(n\) is the number of data points.

06

Confidence Interval

Construct a 95% confidence interval for \(B\) using the formula: \(\widehat{B}\pm t_{\alpha /2, n-2} * \frac{S_{e}}{\sqrt{\Sigma(x_{i}-\bar{x})^{2}}}\), where \(t_{\alpha /2, n-2}\) is the t critical value.

07

Hypothesis Testing

Test at a 1% significance level whether \(B\) and \(\rho\) are different from zero by conducting a two-tailed hypothesis test. The null hypothesis is that the population coefficient (\(B\) and \(\rho\)) equals zero, and the alternate hypothesis is that it does not equal zero. If the computed t-value is less than the t critical value at 1% significance level, fail to reject the null hypothesis. If it is greater, reject the null hypothesis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Correlation Coefficient

The correlation coefficient, denoted as \( r \), is a statistical measure that tells us about the strength and direction of the relationship between two variables. In the context of linear regression, it helps us understand how well the independent variable (such as GPA) can predict the dependent variable (like starting salary). The correlation coefficient can range from -1 to 1.

Here are the interpretations for the values of \( r \):

• \( r = 1 \): Perfect positive correlation, meaning as one variable increases, the other also increases in perfect proportion.
• \( r = -1 \): Perfect negative correlation, indicating that as one variable increases, the other decreases in perfect proportion.
• \( r = 0 \): No correlation, showing that there is no predictable relationship between the variables.

The square of the correlation coefficient, \( r^2 \), is called the coefficient of determination. It tells us the proportion of the variance in the dependent variable that is predictable from the independent variable. For example, an \( r^2 \) value of 0.64 means that 64% of the variance in starting salaries can be explained by GPA.

Standard Deviation

The standard deviation of errors, often denoted as \( S_e \), is crucial in regression analysis. It gives us a measure of the spread of the observed data points around the regression line. Essentially, \( S_e \) provides insight into how much the actual data points deviate from their predicted values based on the regression line.

In formula terms, \( S_e \) is calculated as follows:

• \( S_e = \sqrt{\frac{\Sigma ( y - \widehat{y} )^{2}}{n-2}} \)

Here, \( y \) represents the observed values, \( \widehat{y} \) are the values predicted by the regression line, and \( n \) is the number of observations.

A small \( S_e \) implies that the data points are close to the fitted regression line, indicating a good fit, while a large \( S_e \) suggests the opposite. Understanding \( S_e \) helps assess the accuracy of predictions made by the regression model.

Confidence Interval

A confidence interval gives a range of values that is likely to contain the true parameter of interest, usually with a certain level of confidence (such as 95%). In regression, we often construct confidence intervals for the slope \( B \) of the regression line, to estimate the effect of the independent variable on the dependent variable.

For a 95% confidence interval for \( B \), you use:

• \( \widehat{B} \pm t_{\alpha /2, n-2} \times \frac{S_e}{\sqrt{\Sigma(x_{i}-\bar{x})^{2}}} \)

Here, \( \widehat{B} \) is the estimated regression coefficient, \( t_{\alpha /2, n-2} \) is the t-value from the t-distribution table, \( S_e \) is the standard deviation of errors, and \( \Sigma(x_{i}-\bar{x})^{2} \) helps adjust for the spread of the data.

If the interval includes zero, this suggests that there's no significant effect of the independent variable on the dependent variable. If it does not include zero, we can be more confident that a true effect exists.

Hypothesis Testing

In the context of regression, hypothesis testing is used to determine if there's enough statistical evidence to support a certain hypothesis about the data. For instance, you might want to test whether the slope \( B \) of your regression line is significantly different from zero at a specific significance level (like 1%).

The process involves:

• **Null Hypothesis (\( H_0 \)):** The slope \( B = 0 \), meaning no relationship between the independent and dependent variables.
• **Alternative Hypothesis (\( H_a \)):** The slope \( B eq 0 \), suggesting a relationship exists.

To test these hypotheses, compute a test statistic, which is often a t-value. This is compared to a critical value from the t-distribution, based on your chosen significance level and degrees of freedom. If the calculated t-value exceeds the critical value, you reject the null hypothesis.

Hypothesis testing can also be applied to the correlation coefficient \( \rho \) to determine if it is greater than zero, indicating a positive relationship. Through these tests, we gain insights into the statistical significance of our regression model.