Question

A child psychologist suspects that there is a linear relationship between the amount of calories in the lunches of preschool-age children and the length of time they are able to focus on a given subject. She calculates a regression line where the slope estimates the length of a child focusing (in minutes) with the number of calories consumed at lunch. She calculates a 95% confidence interval for the regression slope to be (-0. 53, -0. 34). Assuming the requirements for this confidence interval are met, what is the correct interpretation

Answer 1

The 95% confidence interval for the regression slope suggests a significant negative linear relationship between calories consumed and focus time in preschool children, indicating a decrease in focus time with an increase in calories.

Step-by-step explanation:

The correct interpretation of the 95% confidence interval for the regression slope (-0.53, -0.34) is that we can be 95% confident that the true slope of the relationship between the number of calories consumed at lunch and the length of time preschool-age children are able to focus falls within this interval. Since the confidence interval does not include zero, this suggests that there is a significant negative linear relationship between the two variables, meaning that as the number of calories increases, the time focused decreases. The slope indicates that for each additional calorie consumed, the time focused is expected to decrease by an amount between 0.53 and 0.34 minutes on average.

Interpretation of the slope: The negative values of the slope in the confidence interval suggest that an increase in the number of calories is associated with a decrease in the amount of time a child can focus. Since confidence intervals are a statistical tool used to estimate the precision of the regression slope estimate, it also tells us about the uncertainty around our estimate. Given that the interval does not contain zero, we reject the null hypothesis that there is no relationship between calories consumed and focus time.

Answer 2

The 95% confidence interval for the regression slope suggests that the true average decrease in children's focus time per additional calorie consumed falls between 0.34 and 0.53 minutes. As calories increase, the focus time decreases.

Step-by-step explanation:

The correct interpretation of the 95% confidence interval for the regression slope of (-0.53, -0.34) is that we are about 95% confident that the true slope of the relationship between the number of calories consumed at lunch and the length of time that preschool-aged children can focus on a given subject falls within this interval. Because the interval consists of negative values, it indicates that as the number of calories increases, the time of focus is estimated to decrease. This interval also suggests that for every additional calorie consumed, the child's focusing time decreases by an amount between 0.34 and 0.53 minutes on average.

The regression equation is used to make predictions about a dependent variable based on the value of an independent variable, often calculated through a process of minimizing the sum of the squares of the vertical distances of the points from the line (least-squares regression). The slope of the regression line is particularly informative, as it describes the average change in the dependent variable for each one-unit increase in the independent variable. Therefore, with the given confidence interval for the slope, it appears that higher calorie intake is associated with a decrease in the time a child stays focused.