STAT177 homework sheet #9

A travel agent wanted to see the relationship between a household’s annual income and how much per week they spent on vacation. From its records, it sampled 8 clients and got the following results (all in thousands of dollars):

Annual income	51.3	69.69	73.58	71.56	56.79	33.48	52.39	41.39
Vacation amount	2.35	3.01	3.58	3.28	2.85	1.81	2.34	1.82

1) Construct a model in which the amount spent on vacation depends on household annual income. State what the model is. (vacation = 0.1878 + 0.0434*income)

2) What percentage of the variation in how much a household spends on vacation is explained by annual household income? (93.55%)

3) Is the model significant? Test at 5%. State the p-value and conclusion. (p-value = 0; conclude the model is significant)

4) If a household earns $75,000 per year, how much would you expect it to spend per week on vacation? Round to the nearest hundred. ($3400)

5) For households that earns $75,000 per year, what is the range of the average amount spent per week on vacation 95% of the time? Round the limits to the nearest hundred. ($3200 to $3700)

6) For a particular household that earns $75,000 per year, what is the range of the amount this household spends on vacation 95% of the time? Round the limits to the nearest hundred. ($2900 to $4000)

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 moderate/high income = 1. Twenty subjects were randomly chosen. These were the results:

grade	study	sleep	gender	income
71	4	6	0	0
75	3	9	0	0
61	6	7	1	0
63	8	6	1	1
63	6	10	0	0
58	7	7	1	0
60	5	9	1	0
90	10	7	0	0
93	9	10	1	0
83	11	8	0	1
73	9	5	0	1
75	12	6	0	0
87	12	5	0	1
88	12	4	1	0
90	14	6	0	0
47	4	4	1	0
98	11	8	1	1
96	11	10	1	0
64	7	7	1	1
45	5	6	1	1

7) Construct a model with grade as the dependent variable and the other variables as the independent variables. State what the model is (grade = 24.3923 + 3.8911*study + 2.9052*sleep – 3.8384*gender – 2.6087*income)

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

9) Is the model significant? (p-value = 0.1%; yes, the model is significant)

10) What individual variables are significant? (study and sleep; these are the only variables in which the p-value is less than 5%)

11) Are there any collinearity problems among the independent variables? (no – all the VIF values are less than 10)

12) Construct a new model using only study and sleep as the independent variables. State what the model is. (grade = 20.0198 + 3.9576*study + 3.0188*sleep)

13) 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)

14) 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)

15) 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)

16) 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)