Which value of r indicates a stronger correlation than +0.40?
· -0.30
· -0.80
· +0.38
· 0
A sales manager for an advertising agency believes there is a relationship between the number of contacts that a salesperson makes and the amount of sales dollars earned. A regression analysis shows the following results.
Assume that I = 33.4 and E(X – 1) = 2814.4. The 95% prediction interval for a particular person making 30 calls is __
· 55.8, 51.5
· 51.4, 55.9
· 46.7, 60.6
· 31.1, 76.2
The Correct Answer and Explanation is :
Answer 1: Stronger Correlation than +0.40
The value that indicates a stronger correlation than +0.40 is -0.80.
Explanation:
- Correlation (r) measures the strength and direction of the relationship between two variables. It ranges from -1 to +1.
- A value closer to 1 or -1 represents a stronger relationship.
- +0.40 indicates a moderate positive correlation.
- -0.80 represents a strong negative correlation, which is stronger than +0.40 in magnitude.
- The other options (-0.30, +0.38, and 0) are weaker in absolute value compared to +0.40.
Answer 2: 95% Prediction Interval for a Salesperson Making 30 Calls
The correct answer is:
(31.1, 76.2).
Explanation:
The 95% prediction interval is used to estimate the range within which an individual prediction will fall, considering variability. It accounts for both the error in estimating the mean and individual deviations from the regression line.
Given data:
- Intercept: -12.201
- Slope (Contacts): 2.195
- X (calls made) = 30
- Mean of X (( \bar{X} )) = 33.4
- Variance in X (E(X – 1)) = 2814.4
- Residual Mean Square Error (MSE) = 86.68
The prediction interval formula is:
[
\hat{Y} \pm t_{critical} \times SE_{prediction}
]
Where:
- ( \hat{Y} = b_0 + b_1X = -12.201 + (2.195 \times 30) = 53.75 )
- ( SE_{prediction} ) accounts for regression standard error and variance.
- The wider range (31.1 to 76.2) correctly represents this uncertainty.
Thus, the 95% prediction interval is (31.1, 76.2), which captures the possible range for an individual making 30 calls.