Question

1. a small - business owner collected data from a random sample on the amount spent on an order and the time it takes to fill the order. the resulting data were used to complete a linear regression analysis of time, in minutes, versus amount spent, in dollars, and to create the following residual plot. based on the residual plot, which condition for inference for the slope of the regression line does not appear to be satisfied? (a) the true relationship between the number of items and the amount spent on an order is linear. (b) for each amount spent on an order, the distribution of time to fill the order is approximately normal. (c) the standard deviation of the distribution of time to fill an order is the same for each amount spent on the order. (d) the data are collected using a random sample or randomized experiment. (e) the sample size is less than 10% of the population size.

Explanation:

Step1: Recall the conditions for inference for the slope of the regression line

The conditions are: linearity (the true relationship is linear), independence (observations are independent), normality (for each \(x\), \(y\) is approximately normal), equal - variance (the standard deviation of \(y\) for each \(x\) is the same), and randomness (data is from a random sample or randomized experiment).

Step2: Analyze the residual plot

A residual plot is used to check the linearity and equal - variance conditions. If the relationship is linear, the residual plot should have a random scatter of points. If the equal - variance (constant standard deviation) condition is met, the spread of the residuals should be approximately the same across all values of the independent variable.
In this residual plot, the spread of the residuals is not constant. For example, the residuals seem to have a larger spread for some values of the amount spent on the order. This violates the equal - variance (constant standard deviation) condition.
The linearity condition is also violated because the residual plot does not show a random scatter. There is a non - random pattern (not just a random cloud of points), which implies that the true relationship between the time to fill the order (dependent variable) and the amount spent on the order (independent variable) is not linear.

Answer:

A. The true relationship between the number of items and the amount spent on an order is linear.