Ok, so I'm trying to do this for my exam review. I get some of it, but not all:
An electric utility is examining the relationship between temperature and electricity use in its service region on warm days. The utility has bivariate data detailing the maximum temperature (denoted by x, in degrees Fahrenheit) and the electricity use (denoted by y, in thousands of kilowatt hours) for a random sample of 28 warm days. For these data, the utility has computed the least-squares regression equation to be yhat= 58.67+3.01x.
Tomorrow's forecast high temperature is 80 degrees Fahrenheit. With this in mind, utility managers have used the regression equation to predict tomorrow's electricity use, but they're also interested in both a prediction interval for the electricity use and a confidence interval for the mean electricity use on days for which the maximum temperature is 80 degrees Fahrenheit. The managers have computed the following for their data:
mean square error (MSE):776.23
some confidence interval expression:.0516
You then have to find the lower and upper limit with a 95% confidence interval.
I know the formula is to find t.05(26)=1.706.
Then its (1.706)(sqrt776.23)(sqrt.0516)
My first problem is, it always .0516, because I thought you added 1 sometimes?
The problem goes on to ask: Consider but do not actually compute the 95% confidence interval for the mean electricity use when the max temp is 80 degrees. how would the confidence interval compare to the prediction interval computed before? I'm lost hear.
Then:For the maximum temp values in this sample, 71 degrees is more extreme than 81 degrees, that is, 71 is farther from the sample mean than 80. How would the 95% confidence interval for the mean electricity use when the maximum temp is 80 degrees compare to the 95% confidence interval for the mean electricity use when the temp is 71 degrees?
I know it looks like alot, but I really need help and there's not even math involved.Thanks so much.
To calculate the lower and upper limits of the 95% confidence interval, you're correct in using the formula t * sqrt(MSE) * sqrt(.0516), where t is the critical value from the t-distribution.
In this case, the critical value is obtained by looking up the t-value for a 95% confidence level with a degrees of freedom of 26 (n-2, where n is the sample size). You correctly found the t-value of 1.706.
Now you can calculate the lower and upper limits by substituting the values into the formula:
Lower limit = 58.67 + (1.706 * sqrt(776.23) * sqrt(.0516))
Upper limit = 58.67 - (1.706 * sqrt(776.23) * sqrt(.0516))
Regarding your question about the ".0516", this seems to be a typo as a confidence interval value cannot be expressed as a decimal. It should be a percentage, typically written as a value between 0 and 1. Double-check the provided information or consult with your instructor for clarification.
Moving on to the comparison between the confidence interval for the mean electricity use and the prediction interval computed earlier, there are some differences to note.
A confidence interval estimates the range within which the population mean (in this case, electricity use) is likely to lie, given the sample data. It provides a fixed range based on the provided confidence level and assumes that the maximum temperature is fixed at 80 degrees Fahrenheit.
On the other hand, a prediction interval estimates the range within which individual observations (in this case, electricity use) are likely to fall, taking into account both the variability within the data and the uncertainty of future values. It provides a wider range than the confidence interval, as it considers the additional uncertainty associated with making predictions for a specific value (in this case, tomorrow's electricity use at 80 degrees Fahrenheit).
Finally, regarding the comparison between the 95% confidence interval for the mean electricity use when the maximum temperature is 80 degrees and when it is 71 degrees, we would expect the interval for 71 degrees to be wider. This is because the further a temperature value is from the sample mean, the greater its impact on the variability of the electricity use. In this case, 71 degrees is more extreme than 80 degrees, implying that the 95% confidence interval for the mean electricity use at 71 degrees will have greater width than the interval for 80 degrees.