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692 :pter 1U: Inference for Regi^B

“real dollars” with constant hu    eąual to that of a dollar in the

year 2000.

Year	1985	1986	1987	1988	1989	1990	1991	1992
Stocks	12.8	34.6	28.8	—23.3	8.3	17.1	50.6	97.0
Bonds	100.8	161.8	10.6	-5.8	-1.4	9.2	74.6	87.1

Year	1993	1994	1995	1996	1997	1998	1999	2000
Stocks Bonds	151.3 84.6	133.6 -72.0	140.1 -6.8	238.2 3.3	243.5 30.0	165.9 79.2	194.3 -6.2	309.0 -48.0

(a)    Make a scatterplot with cash flow into stock funds as the explanatoiy variable. Find the least-squares linę for predicting net bond investments from net stock investments. What do the data suggest?

(b)    Is there statistically significant evidence that there is some straight-line relationship between the flows of cash into bond funds and stock funds? (State hypotheses, give a test statistic and its P-value, and State your conclusion.)

(c)    What faet about the scatterplot explains why the relationship described by the least-sąuares linę is not significant?

10.4 How well does the number of beers a student drinks predict his or her blood alcohol content? Sixteen student volunteers at Ohio State University drank a randomly assigned number of cans of beer. Thirty minutes later, a police officer measured their blood alcohol content (BAC). Here are the data:⁹

Student	1	2	3	4	5	6	7	8
Beers	5	2	9	8	3	7	3	5
BAC	0.10	0.03	0.19	0.12	0.04	0.095	0.07	0.06

Student	9	10	11	12	13	14	15	16
Beers	3	5	4	6	5	7	1	4
BAC	0.02	0.05	0.07	0.10	0.085	0.09	0.01	0.05

The students were eąually divided between men and women and differed in weight and usual drinking habits. Because of this variation, many students don't believe that number of drinks predicts blood alcohol well.

(a) Make a scatterplot of the data. Find the eąuation of the least-sąuares regression linę for predicting blood alcohol from number of beers and add this linę to your plot. What is r² for these data? Briefly summarize what your data analysis shows.

(b) Is there significant evidence that drinking morę beers increases

blood alcohol on the average in the population of all students? State hypotheses, give a test statistic and P-value, and State your conclusion.

10.5 The capacity (bits) of the largest DRAM (dynamie random access memory) chips commonly available at retail has inereased as follows:¹⁰

Year	1971	1980	1987	1993	1999	2000
Bits	1,024	64,000	1,024,000	16,384,000	256,000,000	512,000,000

(a)    Make a scatterplot of the data. Growth is much faster than linear.

(b)    Plot the logarithm of DRAM capacity against year. These points are close to a straight linę.

(c)    Regress the logarithm of DRAM capacity on year. Give a 90% confidence interval for the slope of the population regression linę.

10.6    Your scatterplot in Exercise 10.4 shows one unusual point: student number 3, who drank 9 beers.

(a)    Does student 3 have the largest residual from the fitted linę? (You can use the scatterplot to see this.) Is this observation extreme in the x direction, so that it may be influential?

(b)    Do the regression again, omitting student 3. Add the new regression linę to your scatterplot. Does removing this observation greatly change predicted BAC? Does r² change greatly? Does the P-value of your test change greatly? What do you conclude: did your work in the previous problem depend heavily on this one student?

10.7    Metatarsus adductus (cali it MA) is a tuming in of the front part of the foot that is common in adolescents and usually corrects itself. Hallux abducto valgus (cali it HAV) is a deformation of the big toe that is not common in youth and often reąuires surgery. Perhaps the severity of MA can help predict the severity of HAV. Table 2.3 (page 120) gives data on 38 consecutive patients who came to a medical center for HAV surgery.¹¹ Using X-rays, doctors measured the angle of deformity for both MA and HAV.

They speculated that there is a positive association—morę serious MA is associated with morę serious HAV.

(a)    Make a scatterplot of the data in Table 2.3. (Which is the explanatory variable?)

(b)    Describe the form, direction, and strength of the relationship between MA angle and HAV angle. Are there any elear outliers in your graph?

(c)    Give a statistical model that provides a framework for asking the question of interest for this problem.

(d)    Translate the question of interest into nuli and altemative hypotheses.

(e)    Test these hypotheses and write a short description of the results. Be surę to include the value of the test statistic, the degrees of freedom, the P-value, and a elear statement of what you conclude.