HE NATURAL GAS CONSUMPTION CASE GasCon2 Consider the situation in which a gas co
ID: 3158218 • Letter: H
Question
HE NATURAL GAS CONSUMPTION CASE GasCon2 Consider the situation in which a gas company wishes to predict weekly natural gas consumption for its city. Previously we used the single predictor variable x, average hourly temperature, to predict y, weekly natural gas consumption. We now consider predicting y on the basis of average hourly temperature and a second predictor variable—the chill index. The chill index for a given average hourly temperature expresses the combined effects of all other major weather-related factors that influence natural gas consumption, such as wind velocity, sunlight, cloud cover, and the passage of weather fronts. The chill index is expressed as a whole number between 0 and 30. A weekly chill index near 0 indicates that, given the average hourly temperature during the week, all other major weather-related factors will only slightly increase weekly natural gas consumption. A weekly chill index near 30 indicates that, given the average hourly temperature during the week, other weather-related factors will greatly increase weekly natural gas consumption. The natural gas company has collected data concerning weekly natural gas consumption (y, in MMcF), average hourly temperature (x1, in degrees Fahrenheit), and the chill index (x2) for the last eight weeks. The data are given below and scatter plots of y versus x1 and y versus x2 are given below the data.
y = 0 + 1x1 + 2x2 +
The Natural Gas Consumption Data
Using the Model y = 0 + 1x1 + 2x2 +
• a.Using the Excel solutions find (on the output) b1 and b2, the least squares point estimates of 1 and 2, and report their values. Then interpret b1 and b2.
• b.Calculate a point estimate of the mean natural gas consumption for all weeks that have an average hourly temperature of 40 and a chill index of 10, and a point prediction of the amount of natural gas consumed in a single week that has an average hourly temperature of 40 and a chill index of 10. Find this point estimate (prediction), which is given at the bottom of the MINITAB output, and verify that it equals (within rounding) your calculated value.
Excel outputs of regression analyses of the data sets related to four case studies introduced previously. Above each output we give the regression model and the number of observations, n, used to perform the regression analysis under consideration. Using the appropriate model, sample size n, and output:
1. Report SSE, s2, and s as shown on the output. Calculate s2 from SSE and other numbers.
2. Report the total variation, unexplained variation, and explained variation as shown on the output.
3. Report R2 and R2 as shown on the output. Interpret R2 and R2. Show how R2 has been calculated from R2 and other numbers.
4. Calculate the F(model) statistic by using the explained and unexplained variations (as shown on the output) and other relevant quantities. Find F(model) on the output to check your answer (within rounding).
5. Use the F(model) statistic and the appropriate critical value to test the significance of the linear regression model under consideration by setting equal to .05.
6. Use the F(model) statistic and the appropriate critical value to test the significance of the linear regression model under consideration by setting equal to .01.
7. Find the p-value related to F(model) on the output. Using the p-value, test the significance of the linear regression model by setting = .10, .05, .01, and .001. What do you conclude?
Model: y = 0 + 1x1 + 2x2 + Sample size: n = 8
1. Find bj, and the t statistic for testing H0: j = 0 on the output and report their values. Show how t has been calculated by using bj and
2. Using the t statistic and appropriate critical values, test H0: j = 0 versus Ha: j 0 by setting equal to .05. Which independent variables are significantly related to y in the model with = .05?
3. Using the t statistic and appropriate critical values, test H0: j = 0 versus Ha: j 0 by setting equal to .01. Which independent variables are significantly related to y in the model with = .01?
4. Find the p-value for testing H0: j = 0 versus Ha: j 0 on the output. Using the p-value, determine whether we can reject H0 by setting equal to .10, .05, .01, and .001. What do you conclude about the significance of the independent variables in the model?
5. Calculate the 95 percent confidence interval for j. Discuss one practical application of this interval.
6. Calculate the 99 percent confidence interval for j.
7. • Report (as shown on the computer output) a point estimate of and a 95 percent confidence interval for the mean natural gas consumption for all weeks having an average hourly temperature of 40°F and a chill index of 10.
8. • Report (as shown on the computer output) a point prediction of and a 95 percent prediction interval for the natural gas consumption in a single week that has an average hourly temperature of 40°F and a chill index of 10.
9. • .Suppose that next week the city's average hourly temperature will be 40°F and the city's chill index will be 10. Also, suppose the city's natural gas company will use the point prediction = 10.333 and order 10.333 MMcf of natural gas to be shipped to the city by a pipeline transmission system. The gas company will have to pay a fine to the transmission SYSTEM if the city's actual gas usage y differs from the order of 10.333 MMcf by more than 10.5 percent—that is, is outside of the range [10.333 ± .105(10.333)] = [9.248, 11.418]. Discuss why the 95 percent prediction interval for y, [9.293, 11.374], says that y is likely to be inside the allowable range and thus makes the gas company 95 percent confident that it will avoid paying a fine.
10. • Find 99 percent confidence and prediction intervals for the mean and actual natural gas consumption referred to in parts a and b. Hint: n = 8 and s = .367078. Optional technical note needed.
y x1 x2 12.4 28 18 11.7 28 14 12.4 32.5 24 10.8 39 22 9.4 45.9 8 9.5 57.8 16 8 58.1 1 7.5 62.5 0Explanation / Answer
alpha = 0.05
alpha = 0.01
SUMMARY OUTPUT Regression Statistics Multiple R 0.986727 R Square 0.97363 Adjusted R Square 0.963081 Standard Error 0.367078 Observations 8 ANOVA df SS MS F Significance F Regression 2 24.87502 12.43751 92.30309 0.000112926 Residual 5 0.673732 0.134746 Total 7 25.54875 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept 13.10874 0.855698 15.31935 2.15E-05 10.90909535 15.30838 10.9091 15.30838 x1 -0.09001 0.014077 -6.39423 0.001386 -0.126200879 -0.05383 -0.1262 -0.05383 x2 0.082495 0.022003 3.749337 0.013303 0.025935624 0.139054 0.025936 0.139054Related Questions
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