Hw 7 Answers

Solutions: Confidence Intervals

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1. X ± 1.96σ/ n; i.e., 48 ± 1.96(4/ 10), which is 48 ± 2.48.

2. The sample proportion is p = 15/500 = .03. Confidence interval:

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p ± 1.96

ˆ

p(1 − p)

ˆ

ˆ

= .03 ± 1.96

n

(.03)(1 − .03)

= .03 ± .015.

500

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3. X ± 1.96s/ n, with X = 110, s = 16, and n = 40, gives 110 ± 4.96.

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4. Test customers: X = 227, sX = 135, nX = 850. Control: Y = 163, sY = 85, nY = 700.

Confidence interval

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(X − Y ) ± 1.96

s2

s2

X

+ Y = 164 ± 11.04.

nX

nY

5. Look up tn−1,α/2 with n = 15 and α = 5%; i.e., t14,.025 which is 2.145. For 90% it’s

1.761. A multiplier of 2.624 leaves 1% probability in each tail and thus results in a 98%

confidence level.

6. This is a matched-pairs setting because we are comparing the same companies before and

after the election. The average of the differences is 2.5 and their standard deviation is

9.7. The sample size is 12, so we use the t-multiplier for 11 degrees of freedom and 95%

confidence (right-tail probability 0.025) which is 2.201. Putting the pieces together we

get a confidence interval of

9.5

2.5 ± 2.201 √ = 2.5 ± 6.0.

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Comment: This confidence interval is very wide. The uncertainty (as measured by the

halfwidth of 6.0) is very large compared with the average increas of 2.5 million. Indeed,

we cannot even be confident that the mean changed after the election because the interval

crosses zero. (Note that for this interpretation of the confidence interval we need to view

the 12 companies as a random sample from a larger hypothetical population.)

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7. (a) No incentive: pX = 18/45 = 0.40, nX = 45. With incentive: pY = 12/52 = 0.23,

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nY = 52. Confidence interval:

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pX − pY ± 1.96

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ˆ

pX (1 − pX ) pY (1 − pY )

ˆ

ˆ

ˆ

+

= 0.17 ± 0.18.

nX

nY

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(b) Now we have pX = 0.40, nX = 450, pY = 0.23, nY = 520. The same formula gives a

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confidence interval of 0.17 ± 0.058. We can make two comments comparing (a) and (b).

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First, the confidence interval in...