ppmidterm2s131

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22S:131 Midterm2 8 April 2010 print name_______________________

1) A rangefinder is used to make 11 measurements of the distance between two fixed points.

These measurements are assumed to be a random sample from n(, 2), and result in asample variance s2 = 1.62 meters. Find the p-value in a test of hyp: 2 = 5 vs. alt: 2 < 5.

a) .01 b) .025 c) .05 d) .10 e) .25

2) The power of the = .05 test in (1) against the alternative

2

= 2 is representable as the areaunder a chi-square curve to the left of ____

a) 6.57 b) 7.88 c) 9.85 d) 10.12 e) 14.99

3) A calculus exam is given to 50 students who take a traditional 15 week course, and 75 who take

an intensive 3 week course. In the traditional course, avr = 68 & SD = 6, while in the intensive

course avr = 65 & SD = 10. Find the p-value in a test of the hypothesis that both courses arethe same on average vs. the alternative that the traditional course is better on average.

a) .112 b) .081 c) .054 d) .035 e) .018

4) A box contains one red marble and white marbles. To test hyp: = 7 vs. alt: > 7 we

sample 3 marbles without replacement and reject the hypothesis if none of the selected marbles

is red. Find the power function of this test.

a) 22

b)

1)2(2

+

c)

12+

d)

1e)

2

2


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5) 200 patients with irregular heartbeats were randomly divided into two equal-sized groups.

Each member of the "treatment group" was given 500 mg of a beta-blocker daily for six weeks,

while each member of the "control group" was given a placebo. 68 members of the treatment

group and 48 members of the control group had normal rhythm at the end of the six week period.Find a 95% confidence interval for the difference pt pc between the 2 probabilities of success.

a) (.066, .334) b) (.074, .326) c) (.188, .212) d) (.109, .291) e) (.132, .268)

6) Find the value of the goodness of fit statistc for testing the hypothesis that the 21 observations

0 0 0 0 0 1 1 1 1 1 1 1 2 2 2 2 2 3 5 6 8

are from a Poisson(2) distribution. Use the categories (0,1) (2, 3) (4, 5, )

a) 4.63 b) 3.88 c) 3.24 d) 2.69 e) 1.36

7) Use the data in the table shown to compute the p-value in a test of the hypothesis that

X and Y are independent variables (answer not exact - - select the choice that is CLOSEST).

Y = 0 Y = 1 Y = 2

X = 1 9 23 8

X = 2 21 17 22

a) .01 b) .025 c) .05 d) .10 e) .25


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8) In a random sample of 10 observations from a continuous distribution, what is the

confidence level of the interval ( X(2), X(6) ) as an estimate of the 40th percentile ?

a) 52.7% b) 62.4% c) 68.1% d) 71.5% e) 78.7%

9) Based on a random sample of size 144, we reject the hypothesis that the 80th percentile = 60

in favor of the alternative that the 80th percentile < 60 if X(122) < 60. Use the normal

approximation, with continuity correction, to find the -level of this test.

a) .095 b) .086 c) .075 d) .064 e) .052

10) Consider a random sample of size 5 from a distribution having pdf f(x) = 2x, 0 < x < 1.

Find E(X(4))

a) .755 b) .783 c) .808 d) .842 e) .878

11) Use the runs test to compute the p-value in a test of the hypothesis that two independent random

samples (the X's & the Y's) are from the same distribution.

x x y x x x y y y y

a) .032 b) .054 c) .084 d) .132 e) .167


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22S:131 Midterm2 8 april 2010 print name_______________________

1) A rangefinder is used to make 11 measurements of the distance between two fixed points.

These measurements are assumed to be a random sample from n(, 2), and result in asample variance s2 = 2.43 meters.

Find the p-value in a test of hyp: 2 = 5 vs. alt: 2 < 5.

a) .01 b) .025 c) .05 d) .10 e) .25

2) The power of the = .05 test in (1) against the alternative 2 = 3 is representable as the area

under a chi-square curve to the left of ____

a) 6.57 b) 7.88 c) 9.85 d) 10.12 e) 14.99

3) A calculus exam is given to 50 students who take a traditional 15 week course, and 75

students who take an intensive 3 week course. In the traditional course, avr = 67 & SD = 6,

while in the intensive course avr = 65 & SD = 10. Find the p-value in a test of the hypothesis

that both courses are the same on average vs. the alternative that the traditional course is better

on average.

a) .112 b) .081 c) .054 d) .035 e) .018

4) A box contains one red marble and white marbles. To test hyp: = 7 vs. alt: > 7 we

sample 3 marbles without replacement and reject the hypothesis if none of the selected marbles

is red. Find the power function of this test.

a) 2

2

b) 1

)2( 2

+

c) 1

2

+

d)

1

e)

22


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5) 200 patients with irregular heartbeats were randomly divided into two equal-sized groups.

Each member of the "treatment group" was given 500 mg of a beta-blocker daily for six weeks,

while each member of the "control group" was given a placebo. 78 members of the treatment

group and 58 members of the control group had normal rhythm at the end of the six week period.

Find a 95% confidence interval for the difference pt pc between the two probabilities ofsuccess.

a) (.066, .334) b) (.074, .326) c) (.188, .212) d) (.109, .291) e) (.132, .268)

6) Find the value of the goodness of fit statistc for testing the hypothesis that the 21 observations

0 0 0 0 0 1 1 1 1 1 1 2 2 2 2 2 2 3 5 6 8

are from a Poisson(2) distribution. Use the categories (0,1) (2, 3) (4, 5, )

a) 4.63 b) 3.88 c) 3.24 d) 2.69 e) 1.36

7) Use the data in the table shown to compute the p-value in a test of the hypothesis thatX and Y are independent variables. (answer not exact - - select the choice that is CLOSEST).

Y = 0 Y = 1 Y = 2

X = 1 9 23 8

X = 2 21 17 22

a) .01 b) .025 c) .05 d) .10 e) .25


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8) In a random sample of 10 observations from a continuous distribution, what is the

confidence level of the interval ( X(2), X(6)) as an estimate of the 40th percentile ?

a) 52.7% b) 62.4% c) 68.1% d) 71.5% e) 78.7%

9) Based on a random sample of size 144, we reject the hypothesis that the 80th percentile = 55

in favor of the alternative that the 80th percentile < 55 if X(123) < 55. Find the -level of this test.

(hint: use the normal approximation, with continuity correction)

a) .095 b) .086 c) .075 d) .064 e) .052

10) Consider a random sample of size 5 from a distribution having pdf f(x) = 2x, 0 < x < 1

Find E(X(4))

a) .755 b) .783 c) .808 d) .842 e) .878

11) Use the runs test to compute the p-value in a test of the hypothesis that two independent random

samples (the X's & the Y's) are from the same distribution.

x x y x x x y y y y

a) .032 b) .054 c) .084 d) .132 e) .167


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12) Suppose E(T) = 2 + and E(W) = 5 + 3. Which estimator is unbiased for ?

a) 3T - W b) 2T - W c) T - 2W d) T - 3W e) none of these are unbiased

13) Suppose X1, , X10 ~ U( + 2, 3 + 4). Find an unbiased estimator of based on X .

a)2

2+Xb)

2

3+Xc)

2

3Xd)

2

2Xe)

2

X

14) Find the variance of the estimator in (13)

a)

420

)2(2

+b)

120

)1( 2+c)

12

2

d)

480

)1( 2e)

12

)12( 2+

ppmidterm2s131

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