MGMT2263

Worksheet #6

You are given the following data of shipping distance and the time it takes for a shipment to travel that distance by courier.

Distance in km	825	215	1070	550	480	920	1350	325	670	1215
Time in days	3.5	1	4	2	1	3	4.5	1.5	3	5

1) Construct a model in which the shipping time depends on the distance. State what the model is. (time = 0.1181 + 0.0036*distance)

2) What percentage of the variation in shipping time is explained by the model? (90.05%)

3) Test the hypothesis that the model is significant at the 5% level of significance. (F = 72.3959; conclude model is significant)

4) Construct a 95% confidence interval of the slope coefficient. If this interval were used to test the hypothesis in question 3, why would the same conclusion be reached? (0.0026 < B1 < 0.0046; hypothesized coefficient of zero is not in the interval)

5) In theory, the residuals are normally distributed with a mean of zero and a common variance. What is the estimate of that variance? (0.2304)

6) Test the hypothesis that there is a significant direct relationship between the variables at a 5% level of significance. (t = 8.51; conclude there is a significant direct relationship)

7) If the distance is 500 km, how many days should you expect the shipping time to be? Round to 2 decimals. (1.91 days)

8) If the distance is 500 km, what is the range of the average shipping time for 95% of the time? (1.48 to 2.34 days)

9) For a particular shipment that travels 500 km, what is the range of the shipping time for 95% of the time? (0.72 to 3.1 days)

Suppose you are given the following information:

X = size of home (in thousands of square feet)

Y = home price (in thousands of dollars)

size	1.82	1.59	1.57	1.81	2.01	1.57	1.87	1.82	1.59	1.95
price	173.1	160	164.6	183.5	194.8	166	178.7	181.5	160.5	196.5

10) Construct a model in which the home price depends on the home size (price = 43.4702 + 75.2556*size)

11) What percentage of the variation in price is explained by the model? (88.52%)

12) Test the hypothesis that the model is significant at the 5% level of significance. (F = 61.6842; conclude model is significant)

13) Construct a 95% confidence interval of the slope coefficient. If this interval were used to test the hypothesis in question 12, why would the same conclusion be reached? (53.1597 < B1 < 97.3515; hypothesized coefficient of zero is not in the interval)

14) If a home has 1500 square feet, what would you expect the price to be? Round to the nearest hundred. ($156,400)

15) If the square footage is 1500 square feet, what is the range of the average home price for 95% of the time? Round to the nearest hundred. ($149,600 to $163,100)

16) If a particular home with 1500 square feet is put up for sale, what is the range of the selling price of this home for 95% of the time? Round to the nearest hundred. ($143,400 to $169,300)

An educational researcher wanted to see what factors influenced school grades and examined the average number of hours students studied for a test, average number of hours of sleep per night, gender and household income level. For the purposes of doing regression, male=0 and female=1 and low income = 0 and medium/high income = 1. Twenty subjects were randomly chosen. A model using all variables was constructed. These are the results:

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.8376
R Square	0.7016
Adjusted R Square	0.6220
Standard Error	9.8975
Observations	20
ANOVA
	df	SS	MS	F	Significance F
Regression	4	3454.6016	863.6504	8.8164	0.0007
Residual	15	1469.3984	97.9599
Total	19	4924.0000
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	VIF
Intercept	24.3923	11.9934	2.0338	0.0601	-1.1710	49.9556
study	3.8911	0.7286	5.3403	0.0001	2.3380	5.4441	1.0850
sleep	2.9052	1.2416	2.3399	0.0335	0.2589	5.5515	1.0700
gender	-3.8384	4.5543	-0.8428	0.4126	-13.5456	5.8688	1.0481
income	-2.6087	4.8213	-0.5411	0.5964	-12.8851	7.6676	1.0797

17) Construct a model with grade as the dependent variable.

(grade = 24.3923 + 3.8911*study + 2.9052*sleep – 3.8384*gender – 2.6087*income)

18) What percentage of the variation in grade is explained by the model? (70.16%)

19) Is the model significant? Test at a 5% level of significance. (F = 8.8164; yes, the model is significant)

20) What individual variables are significant at a 5% level of significance? (study and sleep)

21) Are there any collinearity problems among the independent variables? (no)

A second model was constructed using only study and sleep as the independent variables. Here are the results:

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.8247
R Square	0.6802
Adjusted R Square	0.6426
Standard Error	9.6247
Observations	20
ANOVA
	df	SS	MS	F	Significance F
Regression	2	3349.2104	1674.6052	18.0775	6.18997E-05
Residual	17	1574.7896	92.6347
Total	19	4924
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	20.0198	10.9148	1.8342	0.0842	-3.0083	43.0480
study	3.9576	0.6876	5.7560	0.0000	2.5070	5.4083
sleep	3.0188	1.1798	2.5589	0.0203	0.5298	5.5079
X Variable 1	X Variable 2	Predicted Value	Std Error Prediction	Lower 95% Mean	Upper 95% Mean	Lower 95% Predict	Upper 95% Predict
10	7	80.7280	2.4491	75.5609	85.8951	59.7745	101.6814

22) State the new model. (grade = 20.0198 + 3.9576*study + 3.0188*sleep)

23) If someone studies for 10 hours on average and sleeps for 7 hours, what would you expect the person’s grade to be, rounding to the nearest whole number? (81)

24) Based on 10 hours of study and 7 hours of sleep, in what grade range would you expect the average grade to fall 95% of the time, rounding to the nearest whole number? (76 to 86)

25) For an individual who studies for 10 hours and sleeps for 7 hours, in what grade range would you expect the person’s grade to fall 95% of the time? (60 to 100 since the mark cannot be above 100)

26) Using the criteria of adjusted r², ANOVA p-value and t-test p-values (using a 5% level of significance), which of the two models is best? (Model #2)