Name ________________________________________________

QS 381 Introduction to Probability and Statistics – Exam 4 March 8,2002

Gordon Swartzman

Maureen Kennedy

1. (20 pts) Six guinea pigs injected with 0.5 mg of a medication took on the average 15.4 seconds to fall asleep with a standard deviation of 2.2 seconds, while six other guinea pigs injected with 1.5 mg of the same medication took on the average 11.6 seconds to fall asleep with a standard deviation of 2.6 seconds. Assuming that the two samples are independent random samples and that both populations come from a normal distribution, test at the 0.05 level of significance the claim that the increase in dosage reduces the average time it takes a guinea pig to fall asleep. Assume the variances of the two guinea pig populations are equal.

=2.41 (using formula for EQUAL variances)

s_xbar1-xbar2 = 1.39

t = 2.73

t₀ = 1.812

2.73>1.812, reject H₀

2. (25 pts) The following data were obtained in an experiment designed to check whether there is a systematic difference in the weights (in grams) or rocks obtained with two scales:

Rock Specimen	Scale 1	Scale 2
1	12.13	12.17
2	17.56	17.61
3	9.33	9.35
4	11.4	11.42
5	28.62	28.61
6	10.25	10.27
7	23.37	23.42
8	16.27	16.26
9	12.4	12.45
10	24.78	24.75

Assuming that the difference between the respective weights can be looked at as a random sample from a normal population, test at a=0.05 the null hypothesis that the difference between the scales is 0.

Use a PAIRED t-test

Sum(d) = -0.2, Sum(d²) = 0.0014

dbar=-0.02

s_d = 0.0287

H₀: m_d = 0, H_a: m_d ¹ 0

t = -2.20

t₀ = +/-2.262

Fail to reject H₀

3. (35 pts). The table below (the output from an EXCEL spreadsheet) gives the US voting age population (x in millions) and the turnout for federal elections (y in millions) for 8 recent elections. The data can be modeled by the regression equation:

yhat = 0.372598*x + 26.47295

voting age pop	turnout
x	y	x^2	y^2	xy	n=8	yhat	(y-yhat)^2
120.3	73.2	14472.09	5358.24	8805.96		71.29649	3.623353
140.8	77.7	19824.64	6037.29	10940.16		78.93475	1.524604
152.3	81.6	23195.29	6658.56	12427.68		83.21963	2.623186
164.6	86.5	27093.16	7482.25	14237.9		87.80258	1.696717
174.5	92.7	30450.25	8593.29	16176.15		91.4913	1.460953
182.8	91.6	33415.84	8390.56	16744.48		94.58386	8.903447
189.5	104.4	35910.25	10899.36	19783.8		97.08027	53.57843
196.5	96.4	38612.25	9292.96	18942.6		99.68846	10.81395
1321.3	704.1	222973.8	62712.51	118058.7	<-Sums ->	704.0973	84.22464
		r		m	b	se(est)
				0.372598	26.47295

a) (10 pts.) Calculate the regression coefficient r between x and y.

b) (5 pts.) Calculate the coefficient of determination r². Given an interpretation of the result.

c) (10 pts).) Calculate the standard error of the estimate s_e.

d) (10 pts.) When the voting age population is 190 million, construct a 99% prediction interval for the turnout in federal elections.

a) r = 0.942

b) r² = 0.942² = 0.887; 88.7% of the variation in turnout is explained by voting age.

c) s_e = 3.75

d) x₀ = 190; yhat = 97.267; t_c = 3.707; E = 15.57

81.70<yhat<112.84

4. (20 pts) In a 1990 study of 1539 adults, 520 said they used alternative medicines (e.g. homeopathic medicines) in the previous year. In a recent study of 2055 adults 865 said they used alternative medicines in the previous year. Homeopathic doctors claim that use of homeopathic medicine had increased since 1990. At a=0.05 can you accept the claim that the proportion of adults using alternative medicines has increased since 1990?

H₀: p₁³p₂ H_a: p₁<p₂

phat₁ = 0.338; phat₂ = 0.421

pbar = 0.385, qbar=0.615

z_c = -1.645

z = -5.06

reject H₀