Download 11/06/2007

Problem Solving
11/06/2007
The CEO claimed that 90% of the 3000 company employees supported his new policy. To check
his claim, a statistician made a survey by taking an SRS of 300 employees with replacement.
Assume that every selected employee has to check either “support” or “not support” but not both.
1. Identify the population.
2. T/F If what the CEO said is true, the sample proportion approximately follows normal
distribution.
3. T/F If what the CEO said is true and the sample is an SRS without replacement, the sample
proportion approximately follows normal distribution.
4. If the sample is an SRS with replacement and what the CEO said is true, please identify the
normal approximation of the sample proportion distribution.
5. T/F If the statistician tripled the sample size of the SRS, the new mean of the normal
approximation for the sample proportion will be three times the old one.
6. Based on the approximated normal distribution obtained in 4, please compute the 10% and
90% percentiles of the sample proportion.
7. T/F There are 80% chance that the sample proportion of the SRS is between the two
percentiles obtained in 5, if what the CEO said is true.
8. Do you believe that the statistician can independently take 500 SRS and each contains 300
employees? Y/N Why?
9. T/F Suppose what the CEO said is true. If the statistician takes 500 SRS with sample size 300
for each, on average 90% of the 500 sample proportions would be larger than the 10%
percentile obtained in 5.
10. Based on the approximated normal distribution in 4, what is the probability for the sample
proportion to be smaller than 240/300?
11. It turns out that only 240 employees in the SRS support the new policy. Is this result
surprising if the CEO is correct? Do you believe the sample or the CEO or both?
Remark:
If var X   X2 and var Y   Y2 , and X is INDEPENDENT from Y, then

E (aX  b)  aEX  b and var( aX  b)  a 2 var X

E ( X  Y )  EX  EY

var( X  Y )  ( X2   Y2 )

Example: If X ~ Normal ( X ,  X2 ) , Y ~ Normal(Y ,  Y2 ) , and X is independent
from Y, then:
X  Y ~ N o r m (alX  Y ,  X2   Y2 ) and X  Y ~ Normal( X  Y ,  X2   Y2 ) .
9. The electric bulbs of brand A have a mean lifetime of 1500 hours with a standard
deviation of 200 hours, while those of brand B have a mean lifetime of 1300 hours with
a standard deviation of 100 hours. Two random samples of 125 bulbs of each brand
are tested.
(a) Identify the distributions of the two sample means.
(b) what is the probability that the brand A bulb sample will have a mean lifetime which
is at least (i) 160 hours, (ii) 250 hours more than sample mean for the brand B bulb
sample?
T/F (a) 0.7972
T/F (b) 0.0022
10. The variance of the sum obtained in tossing a pair of fair dice is:______________
The variance of the average number obtained in tossing a pair of fair dice is:______
(a) 35 / 12
(b)
35 / 12
(c) 35 / 6
(d) 9 / 12