The managers of an outdoor coffee stand in Coast City are examining the relation
ID: 3315946 • Letter: T
Question
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 y,in dollars) and the maximum temperature (denoted by X, in degrees Fahrenheit for each of 39 randomly selected days during the past year. The least.squares regression equation computed from their data is y = 25 1916-10.99 x. Tommorrow's forecast high is 68 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 onych the maximum temperature is 68. They have computed the following for their data: mean square r(ME 1 39 2336.27 (68-X)-~ 0.0590. , (x-x) where 1, 2, ,x,9 denote temperatures in the sample, and denotes their mean Based on this information, and assuming that the regression assumptions hold, answer the questions in the table below. (If necessary, consult a list of formulas)Explanation / Answer
(a) Prdicted value of coffee sales, y^ when the maximum temeperature is 68 F
y^ = 2519.16 - 10.99 x
y^ = 2519.16 - 10.99 * 68 = $ 1771.84
95% confidence interval = y^ +- tcriticalse * sqrt[1/n + (x - x)2 /SSxx]
Here dF = 39 - 2 = 37 and alpha = 0.05
t37,0.05 = 2.0262
95% confidence interval = y^ +- tcriticalse * sqrt[1/n + (x - x)2 /SSxx]
see = sqrt(MSE) = sqrt (2336.27) = 48.335
Lower limit = 1771.84 - 2.0262 * 48.335 * 0.0590 = $ 1766.06
Upper limit = 1771.84 + 2.0262 * 48.335 * 0.0590 = $ 1777.62
Question 2.
Here the 95% prediction interval for the predicted value of coffee sales will have more width that the confidence interval for the predicted value of coffee sales .
Question 3
Here as the 84 F value is farther from the sample mean maximum temperature. As this term is in the numerator so that will increase the width of confidence interval as compared to the width of confidence iterval for maximum temperature of 68F.
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