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 38 randomly selected days during the past year. The least-squares regression equation computed from their data is y = 2520.46 – 11.04.x. Tommorrow's forecast high is 66 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 66 degrees Fahrenheit. They have computed the following for their data. • mean square error (MSE) = 2624.44 (66 –7)? 1 = 0.0468, wherex, X21 X38 denote temperatures in the sample, and x denotes their mean 38 38 i = 1 Based on this information, and assuming that the regression assumptions hold, answer the questions below. (If necessary, consult a list of formulas.) (a) What is the 95% confidence interval for the mean coffee sales (in dollars) when the maximum temperature is 66 degrees Fahrenheit? (Carry your intermediate computations to at least four decimal places, and round your answer to at least one decimal place.) Lower limit:| Upper limit: (b) Consider (but do not actually compute) the 95% prediction interval for an individual value for coffee sales when the maximum temperature is 66 degrees Fahrenheit. How would the confidence interval computed above compare to this prediction interval (assuming that both intervals are computed from the same sample data)? The confidence interval would be positioned to the right of the prediction interval. The confidence interval would be identical to the prediction interval. The confidence interval would have the same center as, but would be wider than, the prediction interval. The confidence interval would be positioned to the left of the prediction interval. The confidence interval would have the same center as, but would be narrower than, the prediction interval.
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 38 randomly selected days during the past year. The least-squares regression equation computed from their data is y = 2520.46 – 11.04.x. Tommorrow's forecast high is 66 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 66 degrees Fahrenheit. They have computed the following for their data. • mean square error (MSE) = 2624.44 (66 –7)? 1 = 0.0468, wherex, X21 X38 denote temperatures in the sample, and x denotes their mean 38 38 i = 1 Based on this information, and assuming that the regression assumptions hold, answer the questions below. (If necessary, consult a list of formulas.) (a) What is the 95% confidence interval for the mean coffee sales (in dollars) when the maximum temperature is 66 degrees Fahrenheit? (Carry your intermediate computations to at least four decimal places, and round your answer to at least one decimal place.) Lower limit:| Upper limit: (b) Consider (but do not actually compute) the 95% prediction interval for an individual value for coffee sales when the maximum temperature is 66 degrees Fahrenheit. How would the confidence interval computed above compare to this prediction interval (assuming that both intervals are computed from the same sample data)? The confidence interval would be positioned to the right of the prediction interval. The confidence interval would be identical to the prediction interval. The confidence interval would have the same center as, but would be wider than, the prediction interval. The confidence interval would be positioned to the left of the prediction interval. The confidence interval would have the same center as, but would be narrower than, the prediction interval.
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Question
Transcribed Image Text: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 38 randomly selected days during the past year. The least-squares regression equation computed from their data is
y = 2520.46 – 11.04.x.
Tommorrow's forecast high is 66 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 66 degrees Fahrenheit. They have computed the following for their data.
• mean square error (MSE) = 2624.44
2
(66-x)
1
z 0.0468, wherex¡, x2,
X3g denote temperatures in the sample, and x denotes their mean
38
38
1'
2
i = 1
Based on this information, and assuming that the regression assumptions hold, answer the questions below.
(If necessary, consult a list of formulas.)
(a) What is the 95% confidence interval for the mean coffee sales (in dollars) when the maximum temperature
is 66 degrees Fahrenheit? (Carry your intermediate computations to at least four decimal places, and round
your answer to at least one decimal place.)
Lower limit:
Upper limit:
(b) Consider (but do not actually compute) the 95% prediction interval for an individual value for coffee sales
when the maximum temperature is 66 degrees Fahrenheit. How would the confidence interval computed
above compare to this prediction interval (assuming that both intervals are computed from the same sample
data)?
The confidence interval would be positioned to the right of the prediction interval.
The confidence interval would be identical to the prediction interval.
The confidence interval would have the same center as, but would be wider than, the prediction
interval.
The confidence interval would be positioned to the left of the prediction interval.
The confidence interval would have the same center as, but would be narrower than, the prediction
interval.
Transcribed Image Text:(c) For the maximum temperature values in this sample, 50 degrees Fahrenheit is more extreme than 66
degrees Fahrenheit is, that is, 50 is farther from the sample mean maximum temperature than 66 is. How
would the 95% confidence interval for the mean coffee sales when the maximum temperature is 50 degrees
Fahrenheit compare to the 95% confidence interval for the mean coffee sales when the maximum
temperature is 66 degrees Fahrenheit?
The interval computed from a maximum temperature of 50 degrees Fahrenheit would be narrower but
have the same center.
O The interval computed from a maximum temperature of 50 degrees Fahrenheit would be wider and have
a different center.
The intervals would be identical.
The interval computed from a maximum temperature of 50 degrees Fahrenheit would be narrower and
have a different center.
The interval computed from a maximum temperature of 50 degrees Fahrenheit would be wider but have
the same center.
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