Question

IC	Price	Income	Temp	Lag-temp
0.386	0.27	78	41	56
0.374	0.282	79	56	63
0.393	0.277	81	63	68
0.425	0.28	80	68	69
0.406	0.272	76	69	65
0.344	0.262	78	65	61
0.327	0.275	82	61	47
0.288	0.267	79	47	32
0.269	0.265	76	32	24
0.256	0.277	79	24	28
0.286	0.282	82	28	26
0.298	0.27	85	26	32
0.329	0.272	86	32	40
0.318	0.287	83	40	55
0.381	0.277	84	55	63
0.381	0.287	82	63	72
0.47	0.28	80	72	72
0.443	0.277	78	72	67
0.386	0.277	84	67	60
0.342	0.277	86	60	44
0.319	0.292	85	44	40
0.307	0.287	87	40	32
0.284	0.277	94	32	27
0.326	0.285	92	27	28
0.309	0.282	95	28	33
0.359	0.265	96	33	41
0.376	0.265	94	41	52
0.416	0.265	96	52	64
0.437	0.268	91	64	71

1. Define the first-order linear model.

a. E(Y) =   β0 +   β1X1 +  β2X2 +  β3X3 + β4X4

b. E(Y)      =          β1X1 +  β2X2 +  β3X3 + β4X4 + e

c. E(Y)      =   β0 +   β1X1 + e

d. None of the above

2. Now fit the model to the data (i.e. run the regression). [I] What proportion of the total variation in ice cream consumption is explained by the regression? [II] What is the name of the statistic used to obtain this information?

a. About 92%; F statistic

b. About 85%; Mean square regression

c. About 1.89%; Coefficient of determination

d. About 85%; Coefficient of determination

e. None of the above

3. Define the null hypothesis for the regression model. What test of significance is used to test this hypothesis?

a. H0: β1 = β2 = β3 = β4 = 0; F

b. H0: β0 = β1 = β2 = β3 = β4 = 0;   F

c. H0: β1 = β2 = β3 = β4 = 0;   t

d. At least one of the coefficients is not equal to zero; F

e. None of the above

4. What is your conclusion, and the basis for your conclusion?

a. Reject H0. Regression is significant; p-value > 0.01

b. Do not reject H0. Regression is NOT significant; p-value < 0.01

c. Do not reject H0. Regression is significant; p-value < 0.01

d. Reject H0. Regression is significant; p-value < 0.01

e. None of the above

5. Upon what basis would this model be useful in predicting ice cream consumption?

a. If R-square is reasonably high and the t values are significant

b. If R-square is reasonable high

c. If F is significant and R-square is reasonably high

d. If F is significant, regardless of the size of R-square

e. If mean square regression is 100 percent

f. None of the above

6. If the model is statistically significant, then how much ice cream consumption is predicted if X1 = 0.35, X2 = 90, X3 = 50, and X4 = 50? To avoid rounding error, it is better to perform your calculations on spreadsheet.

a. 0.2580

b. 0.3429

c. 0.3578

d. None of the above

7. Which of the following is a measure of unexplained variation in the study? What is its value?

a. Standard error of the estimate; 0.0235

b. Mean square error; 0.0235

c. Sum of squares error; 0.0005

d. None of the above

8. Is there evidence that family income has an impact on the amount of ice cream consumed?

a. Yes. At the 0.05 level of significance

b. No. Some of the t values are not significant

c. No. Although the F statistic is significant, not all the t values are significant

d. Yes. The time variable is significant

e. None of the above

9. State the hypothesis to test if the price of ice cream contributes information in the prediction of ice cream consumption

a. H0: β1 = 0

b. H0: β2 = 0

c. H0: β3 = 0

d. H0: β1 = β2 = β3 = β4 = 0

e. None of the above

10. Is there evidence to suggest that price of ice cream has an impact on the amount of ice cream consumed?

a. Yes, at the 0.05 level of significance

b. No, the t value is not significant although it does seem that consumption goes down with price

c. No, the t value is not significant although it does seem that consumption increases with price

d. No, although the F statistic is significant

e. None of the above

Answer 1

Using excel, running a linear regression for the given data,

Home InsertPage Layout Formulas Review View Clear Connections Properties AI AZ 솬습 From From From From Other Existing Refresh Access Web Text Sources Connections All Sort Filer Reapply Text to Remove Data C Edit Links Connections Advanced Columns Duplicates Validation Get External Data Sort&Filter Data Tools J17 0.386 027 0.37 0.282 0.393 0.2 Price Income Temp Lag-temp 56 63 68 69 65 41 78 79 81 80 76 Data Analysis 77 63 Analysis Tools 0.425 0.28 OK 5 0.406 0.272 70.3440.262786561F-Test Two-Sample for Variances Covariance Descriptive Statistics Exponential Smoothing 69 Fourier Analysis Histogramm Moving Average Random Number Generation Rank and Percentile Regressiorn 8 0327 0.275 61 47 32 24 47 32 24 0288 0.267 79 0 0.2690.265 1 0.256 0.277 0.286 0.282 0.298 0.27 4 0.3290.272 5 0.3180.287 0.277 79 85 86 83 26 32 AC) 32 AX) 0.381 63

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The output is obtained as:

SUMMARY OUTPUT Regression Statistics MultipleR R Square Adjusted R Square Standard Error Observations 0.923 0.851 0.826 0.023 29 ANOVA MS Significance F Regression Residual Total 4 0.076 0.013 0.089 0.01934.341 0.001 0.000 28 Coefficients Standard Error tstat P-value Lower 95% Upper 95% Intercept Price Income Temp Lag-temp 0.175 0.558 0.547 0.0012.265 0.0011.065 0.001 0.236 1.457 0.000 0.001 0.002 0.714 0.305 0.002 0.001 0.003 0.482 0.589 0.033 0.298 0.000 0.485 0.847 0.003 0.002 0.004 5.025

1. The model has 4 predictors.Hence the first order linear model is given by:

b. E(Y) = β1X1 +  β2X2 +  β3X3 + β4X4 + e

where e = residual/ error

The fitted model is given by

2. The coefficient of determination of the fitted model gives the proportion of variation in the dependent variable that is explained by the explanatory variables.

From the regression output,

SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 0.923 0.851 0.826 0.023 29

The name and value of the statistic used to obtain this information is d. About 85%; Coefficient of determination

3. The null hypothesis for regression model is given by:

a. H0: β1 = β2 = β3 = β4 = 0; F

4.

ANOVA df F Significance F 0.000 MS Regression Residual Total 4 24 28 0.076 0.019 34.341 0.013 0.001 0.089 O.ixn

d. Reject H₀: Regression is significant; p-value < 0.01

5. This model be useful in predicting ice cream consumption:

c. If F is significant and R-square is reasonably high

6. Substituting X1 = 0.35, X2 = 90, X3 = 50, and X4 = 50 in the fitted model:

E(Y) = 0.125 - 0.305 X₁ + 0.002 X₂ + 0.001 X₃ + 0.003 X₄

= -0.125 -0.305 (0.35) + 0.002 (90) + 0.001 (50) + 0.003 (50)

= 0.3429

7.The measure of Unexplained variation is:

a. Standard error of the estimate; 0.0235

8. To test whether family income has an impact on the amount of ice cream consumed using t test,

Coefficients Standard Error tstat P-value Lower 95% Upper 95% Intercept Price 0.236 1.457 0.125 0.175 0.558-0.547 0.714 0.482 0.485 -0.305 0.589 0.847 Income0,002 0.00 2265 0.033 0.000003 Temp Lag-temp 0.001 0.003 1.065 5.025 0.001 0.298 0.001 0.002 0.002 0.001 0.000 0.004

a. Yes. At the 0.05 level of significance.

(Since the p-value of the t test = 0.033 < 0.05)

9. The hypothesis to test if the price of ice cream contributes information in the prediction of ice cream consumption:

a. H0: β1 = 0

(Since the slope coefficient corresponding to the predictor 'Price' is  β1.And slope measures the change in IC for a unit change in price)

10. To test whether price of ice cream has an impact on the amount of ice cream consumed,

Coefficients Standard Error 0.175 0.558 0.001 tstat P-value Lower 95% Upper 95% Intercept Price Income Temp Lag-temp 0.125 0.305 0.002 0.001 0.003 0.714 0.547 2.265 0.0011.065 0.298 5.025 0.482 0.589 0.033 0.236 1.457 0.000 0.001 0.002 0.485 0.847 0.003 0.002 0.004 0.001 0.000

b. No, the t value is not significant although it does seem that consumption goes down with price

(Since, the p-value corresponding to the test statistic 0.589 > 0.05,and the slope coefficient is negative (-0.305), as the price increases, the IC decreases.)