A small business tracks how advertising spending relates to weekly sales.
$\left(C\right)$ Create a prediction interval for the value in part $\left(B\right)$.

• A. $(9016.22,11383.78)$
• B. $(8755.90,11644.10)$
• C. $(9066.62,11434.18)$
• D. $(8806.30,11694.50)$

Answer: (C)

Verified step by step guidance

1. Step 1: Understand the problem context. We have data on weekly advertising spending and corresponding sales for 15 weeks. The goal is to create a prediction interval for a specific advertising spending value (from part B, presumably a given advertising amount).
2. Step 2: Fit a simple linear regression model where Sales (dependent variable) is predicted by Advertising spending (independent variable). This involves calculating the regression line equation: \(\hat{y} = b_0 + b_1 x\), where \(b_0\) is the intercept and \(b_1\) is the slope.
3. Step 3: Calculate the predicted sales value \(\hat{y}\) for the given advertising spending value from part B using the regression equation.
4. Step 4: Compute the standard error of the prediction, which accounts for both the variability in the estimate of the mean response and the variability of individual observations around the regression line. The formula for the prediction interval standard error is: \(SE_{pred} = s \sqrt{1 + \frac{1}{n} + \frac{(x_0 - \bar{x})^2}{\sum (x_i - \bar{x})^2}}\) where \(s\) is the standard error of the regression, \(n\) is the number of observations, \(x_0\) is the given advertising value, and \(\bar{x}\) is the mean of the advertising values.
5. Step 5: Determine the critical t-value from the t-distribution with \(n-2\) degrees of freedom for the desired confidence level (usually 95%). Then, construct the prediction interval as: \(\hat{y} \pm t_{\alpha/2, n-2} \times SE_{pred}\) This interval gives a range where we expect the actual sales to fall for the given advertising spending.