Question

For the X,Y data below, compute: a. \(\mathrm{r}\) and determine if it is significantly different from zero. b. the slope of the regression line and test if it differs significantly from zero. c. the \(95 \%\) confidence interval for the slope. $$ \begin{array}{|l|l|} \hline X & Y \\ \hline 4 & 6 \\ \hline 3 & 7 \\ \hline 5 & 12 \\ \hline 11 & 17 \\ \hline 10 & 9 \\ \hline 14 & 21 \\ \hline \end{array} $$

Short answer

Correlation is significant. The slope is significant. Confidence interval does not include zero.

Step-by-step solution

Calculate the Mean

First, calculate the mean of the X values and the Y values. Add up all X values: 4, 3, 5, 11, 10, 14; and divide by the number of observations (6). Do the same for Y values: 6, 7, 12, 17, 9, 21.

Calculate the Correlation Coefficient (r)

Use the formula for correlation coefficient: \[ r = \frac{n(\sum XY) - (\sum X)(\sum Y)}{\sqrt{[n\sum X^2 - (\sum X)^2][n\sum Y^2 - (\sum Y)^2]}}\]Substitute the calculated values of \(\sum X\), \(\sum Y\), \(\sum XY\), \(\sum X^2\), and \(\sum Y^2\) to find \(r\).

Significance Test for Correlation

To test if \(r\) is significantly different from zero, use the t-test statistic:\[t = \frac{r\sqrt{n-2}}{\sqrt{1-r^2}}\]Compare the calculated t-value with the critical t-value at \(n-2\) degrees of freedom for a desired confidence level, like 95%.

Calculate the Slope of the Regression Line

The slope \(b\) of the regression line is calculated using:\[b = \frac{n(\sum XY) - (\sum X)(\sum Y)}{n\sum X^2 - (\sum X)^2}\]Use the previously calculated sums in this formula to determine \(b\).

Test Significance of the Slope

To test if the slope differs significantly from zero, use the t-test with:\[t = \frac{b}{SE_b}\]where \(SE_b = \frac{s}{\sqrt{\sum (X - \bar{X})^2}}\) and \(s\) is the standard error of the estimate. Compare this t-value to the critical t-value for \(n-2\) degrees of freedom.

Calculate the Confidence Interval for the Slope

Compute the 95% confidence interval for the slope using:\[CI = b \pm t_{critical} \times SE_b\]where \(t_{critical}\) is the critical t-value for \(n-2\) degrees of freedom.

Key concepts

Correlation Coefficient

The correlation coefficient, often represented by the symbol \(r\), is a numerical measure that describes the strength and direction of a linear relationship between two variables, X and Y. Its value ranges from -1 to 1:

• A value of 1 signifies a perfect positive linear relationship, meaning as X increases, Y increases proportionally.


• A value of -1 indicates a perfect negative linear relationship, implying that as X increases, Y decreases proportionally.


• A coefficient of 0 suggests no linear relationship between the variables.

To compute \(r\), you use the formula:\[ r = \frac{n(\sum XY) - (\sum X)(\sum Y)}{\sqrt{[n\sum X^2 - (\sum X)^2][n\sum Y^2 - (\sum Y)^2]}} \]where \(n\) is the number of data points, and the summations run over each corresponding product of the dataset values. Calculating \(r\) helps us understand if a linear regression model is appropriate.

Significance Test

A significance test helps to determine if the calculated correlation coefficient \(r\) significantly differs from zero, indicating a meaningful relationship between variables. A common approach is using a t-test. The t-test formula for correlation is:\[ t = \frac{r\sqrt{n-2}}{\sqrt{1-r^2}} \]This formula uses \(n-2\) degrees of freedom, where \(n\) is the number of data points. The calculated t-value is then compared against a critical t-value from the t-distribution table, basing on a chosen significance level (often 95%).

• If the t-value exceeds the critical t-value, \(r\) is significantly different from 0.


• This suggests that there is significant evidence of a linear relationship.

Significance testing confirms the reliability of \(r\) as a measure of association.

Confidence Interval

A confidence interval provides a range of values, derived from the sample data, in which the true population parameter is expected to lie. For a regression slope \(b\), a 95% confidence interval implies that if the sample were repeated numerous times, the interval would capture the true slope 95% of the time. The confidence interval for the regression slope is calculated as follows:\[ CI = b \pm t_{critical} \times SE_b \]Here, \(b\) is the slope, \(t_{critical}\) is the critical t-value for \( n-2 \) degrees of freedom, and \( SE_b \) is the standard error of the slope. This interval helps us understand the precision of \(b\) and provides insights into how the relationship could vary with different samples.

Regression Line

A regression line represents the best fit line through a set of data points in a scatterplot, modeling the relationship between the independent variable \(X\) and the dependent variable \(Y\). The equation of the regression line is:\[ Y = a + bX \]where \(a\) is the y-intercept and \(b\) is the slope of the line. The slope \(b\) is crucial as it indicates the amount of change in \(Y\) for a one-unit change in \(X\). It is calculated using:\[ b = \frac{n(\sum XY) - (\sum X)(\sum Y)}{n\sum X^2 - (\sum X)^2} \]The regression line provides a way to predict \(Y\) based on known values of \(X\), making it invaluable in forecasting and decision-making.

T-Test

A t-test is used in regression analysis to check if the slope of the regression line is significantly different from zero. This process helps determine whether the independent variable \(X\) has a statistically significant impact on the dependent variable \(Y\). The t-test statistic for the slope is calculated as:\[ t = \frac{b}{SE_b} \]where \(b\) is the slope, and \(SE_b\) is the standard error of the slope. This test evaluates the null hypothesis that the slope (\(b\)) is equal to zero.

• If the computed t-value is greater than the critical value from the t-distribution for \( n-2 \) degrees of freedom, we reject the null hypothesis.


• Thus, concluding that \(X\) has a significant impact on \(Y\).

Using a t-test ensures the robustness and reliability of our regression findings.