Sales Performance. In exercise 7, the data on y= annual...

Sales Performance. In exercise $7,$ the data on $y=$ annual sales (\$ 1000 s) for new customer accounts and $x=$ number of years of experience for a sample of 10 salespersons provided the estimated regression equation $\hat{y}=80+4 x .$ For these data $\bar{x}=7$ $\Sigma\left(x_{i}-\bar{x}\right)^{2}=142,$ and $s=4.6098$ a. Develop a $95 \%$ confidence interval for the mean annual sales for all salespersons with nine years of experience. b. The company is considering hiring Tom Smart, a salesperson with nine years of experience. Develop a $95 \%$ prediction interval of annual sales for Tom Smart. c. Discuss the differences in your answers to parts (a) and (b).

Sales Performance. In exercise $7,$ the data on $y=$ annual sales (\$ 1000 s) for new customer accounts and $x=$ number of years of experience for a sample of 10 salespersons provided the estimated regression equation $\hat{y}=80+4 x .$ For these data $\bar{x}=7$ $\Sigma\left(x_{i}-\bar{x}\right)^{2}=142,$ and $s=4.6098$
a. Develop a $95 \%$ confidence interval for the mean annual sales for all salespersons with nine years of experience.
b. The company is considering hiring Tom Smart, a salesperson with nine years of experience. Develop a $95 \%$ prediction interval of annual sales for Tom Smart.
c. Discuss the differences in your answers to parts (a) and (b).

Key Concepts

Linear Regression

Linear regression is a statistical method that models the relationship between a dependent variable and one or more independent variables by fitting a linear equation to observed data. It helps to predict the value of the response variable based on the predictors, and it is foundational for many data analysis techniques.

Confidence Interval for the Mean Response

This interval estimates the range within which the true mean response is likely to fall for a given value of the predictor. In the context of regression, it provides the uncertainty associated with the estimated mean outcome for a specified level of the predictor variable, incorporating both the variability in the data and the precision of the estimated regression line.

Prediction Interval for a New Observation

A prediction interval is used to estimate the range within which a single new observation is likely to fall, given a specific predictor value. It accounts not only for the uncertainty in estimating the mean response (as in the confidence interval) but also for the additional variability inherent in individual observations, which generally makes it wider than the confidence interval.

Comparison of Confidence and Prediction Intervals

The key difference between a confidence interval for the mean response and a prediction interval for an individual outcome is that the prediction interval includes both the uncertainty in the estimated mean (sampling variability) and the random error of an individual observation. This leads to a wider interval for predicting a single future observation compared to estimating the mean outcome for given predictor values.

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