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Regression Analysis

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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 …show more content…

The next term in the prediction interval formula is the standard error of the estimate, [pic]. It can be computed from the mean square error (MSE), which is given to be [pic]:

[pic].

The last part of the prediction interval formula consists of the square root of the sum of [pic] and a fairly long expression. We do not need to compute the long expression, though, because we were given its value: [pic]. We have

With this information, we can compute the [pic] prediction interval for the coffee sales given a maximum temperature of [pic] degrees Fahrenheit:

[pic].

Upon simplification, this is the interval whose lower limit is approximately [pic] and whose upper limit is approximately [pic]

2. Because there 's more precision involved in estimating the mean of a distribution than in predicting a particular observation from that distribution, we would expect the confidence interval to be narrower than the prediction interval. We can verify this by comparing the formulas for computing the intervals (shown near the top). As noted previously, the only difference between the prediction interval formula and the confidence interval formula is that the prediction interval formula has a [pic] in the sum underneath the square root, while the confidence interval formula does not. This makes the margin of error (the term following the "[pic]") greater in the prediction interval formula than in the confidence interval formula, which means that the

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