Confidence Intervals and Prediction Intervals from Simple Linear Regression
Confidence intervals and prediction intervals from simple linear regression
The managers of an outdoor coffee stand in Coast City are examining the relationship between coffee sales and daily temperature. They have bivariate data detailing the stand 's coffee sales (denoted by [pic], in dollars) and the maximum temperature (denoted by [pic], in degrees Fahrenheit) for each of [pic] randomly selected days during the past year. The least-squares regression equation computed from their data is [pic].
Tommorrow 's forecast high is [pic] degrees Fahrenheit. The managers have used the regression equation to predict the stand 's coffee sales for tomorrow. They now are interested in both a prediction interval for tomorrow 's coffee sales and a confidence interval for the mean coffee sales on days on which the maximum temperature is [pic]. They have computed the following for their data:
• mean square error (MSE) [pic] [pic];
•
where [pic] denote the theater revenues in the sample, and [pic] denotes their mean.
The least-squares regression equation can be used to predict the value of one variable (called the dependent variable, often denoted by [pic]) based on a given value of the other variable (called the independent variable, often denoted by [pic]). When we make such a prediction, it is useful to obtain a prediction interval for an individual value of [pic] given a value of [pic]. For example, a [pic] prediction interval for an individual value of [pic], given that [pic], is an interval that is constructed by a method that will capture the actual value of [pic] (when [pic]) about [pic] of the time. In addition, it can be useful to obtain a confidence interval for the mean of the distribution of [pic] given a value of [pic]. For example, a [pic] confidence interval for the mean value of [pic], given that [pic], is an interval that is constructed by a method that will capture the mean of the distribution of [pic] (when
The managers of an outdoor coffee stand in Coast City are examining the relationship between coffee sales and daily temperature. They have bivariate data detailing the stand 's coffee sales (denoted by [pic], in dollars) and the maximum temperature (denoted by [pic], in degrees Fahrenheit) for each of [pic] randomly selected days during the past year. The least-squares regression equation computed from their data is [pic].
Tommorrow 's forecast high is [pic] degrees Fahrenheit. The managers have used the regression equation to predict the stand 's coffee sales for tomorrow. They now are interested in both a prediction interval for tomorrow 's coffee sales and a confidence interval for the mean coffee sales on days on which the maximum temperature is [pic]. They have computed the following for their data:
• mean square error (MSE) [pic] [pic];
•
where [pic] denote the theater revenues in the sample, and [pic] denotes their mean.
The least-squares regression equation can be used to predict the value of one variable (called the dependent variable, often denoted by [pic]) based on a given value of the other variable (called the independent variable, often denoted by [pic]). When we make such a prediction, it is useful to obtain a prediction interval for an individual value of [pic] given a value of [pic]. For example, a [pic] prediction interval for an individual value of [pic], given that [pic], is an interval that is constructed by a method that will capture the actual value of [pic] (when [pic]) about [pic] of the time. In addition, it can be useful to obtain a confidence interval for the mean of the distribution of [pic] given a value of [pic]. For example, a [pic] confidence interval for the mean value of [pic], given that [pic], is an interval that is constructed by a method that will capture the mean of the distribution of [pic] (when