[DATA] Crickets make a chirping noise by sliding their wings rapidly over each other. Perhaps you have noticed that the number of chirps seems to increase with the temperature. The following table lists the temperature (in degrees Fahrenheit, °F) and the number of chirps per second for the striped ground cricket.

f. If the residuals are normally distributed, construct a 95% confidence interval for the slope of the true least-squares regression line.

Verified step by step guidance

1. Step 1: Calculate the least-squares regression line for the data, which has the form \(\hat{y} = b_0 + b_1 x\), where \(b_1\) is the slope and \(b_0\) is the intercept. You will need to find \(b_1\) (the slope) and \(b_0\) using the formulas: \[b_1 = \frac{S_{xy}}{S_{xx}} = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}\] \[b_0 = \bar{y} - b_1 \bar{x}\] where \(\bar{x}\) and \(\bar{y}\) are the sample means of \(x\) and \(y\) respectively.
2. Step 2: Calculate the residuals for each data point, which are the differences between the observed values \(y_i\) and the predicted values \(\hat{y}_i\) from the regression line: \[e_i = y_i - \hat{y}_i\] Then compute the residual standard error (also called the standard error of the estimate), \(s\), using: \[s = \sqrt{\frac{\sum e_i^2}{n - 2}}\] where \(n\) is the number of data points.
3. Step 3: Calculate the standard error of the slope \(b_1\), denoted \(SE_{b_1}\), using the formula: \[SE_{b_1} = \frac{s}{\sqrt{\sum (x_i - \bar{x})^2}}\] This measures the variability of the slope estimate.
4. Step 4: Determine the critical value \(t^*\) from the \(t\)-distribution for a 95% confidence interval with \(n - 2\) degrees of freedom. This value corresponds to the two-tailed probability of 0.05.
5. Step 5: Construct the 95% confidence interval for the slope \(\beta_1\) using the formula: \[b_1 \pm t^* \times SE_{b_1}\] This interval estimates the range of plausible values for the true slope of the regression line, assuming the residuals are normally distributed.

Key Concepts

Least-Squares Regression Line

The least-squares regression line is a straight line that best fits the data by minimizing the sum of the squared differences between observed and predicted values. It models the relationship between an explanatory variable (temperature) and a response variable (chirps per second). The slope indicates how much the response variable changes for a one-unit increase in the explanatory variable.

Residuals and Their Normality

Residuals are the differences between observed values and the values predicted by the regression line. Assuming residuals are normally distributed is important for making valid inferences, such as constructing confidence intervals or conducting hypothesis tests about the regression parameters.

Confidence Interval for the Slope

A confidence interval for the slope estimates the range of plausible values for the true slope of the regression line with a given level of confidence (e.g., 95%). It is calculated using the estimated slope, its standard error, and the appropriate t-distribution critical value, assuming normality of residuals.