Download math20812: practical statistics i semester 2 problems on sampling

MATH20812: PRACTICAL STATISTICS I
SEMESTER 2
PROBLEMS ON SAMPLING DISTRIBUTIONS
1. An optical firm purchases glass to be ground into lenses, and it is known from past experience
that the variance of the refractive index of this kind of glass is 1.26 × 10−4 . As it is important
that the various pieces of glass have nearly the same index of refraction, the firm rejects such
a shipment if the sample variance of 20 pieces selected at random exceeds 2 × 10−4 . Assuming
that the sample values may be looked upon as a random sample from a normal population,
what is the probability that a shipment will be rejected?
2. A manufacturer of fuses claims that with a 20% overload, the fuses will blow in 12.4 minutes
on the average. To test this claim, a sample of 20 of the fuses was subjected to a 20% overload,
and the times it took them to blow had an average of 10.63 minutes and a standard deviation
of 2.48 minutes. If it can be assumed that the data constitute a random sample from a normal
distribution, do they tend to support or refute the manufacturer’s claim?
3. If two independent random samples of size n1 = 7 and n2 = 13 are taken from a normal
population, what is the probability that the variance of the first sample will be at least as
large as of the second sample?
4. A manufacturer of car batteries guarantees that his batteries will last, on the average, 3 years
with a standard deviation of 1 year. If five of these batteries have lifetimes of 1.9, 2.4, 3, 3.5
and 4.2 years, is the manufacturer still convinced that his batteries have a standard deviation
of 1 year?
5. If a cigarette manufacturer claims that his cigarettes have an average nicotine content of 18.3
milligrams, is it likely that we could select a random sample of eight cigarettes and find the
nicotine contents to be 20, 17, 21, 19, 22, 21, 20 and 16 milligrams?
6. For an F distribution find Fν1 ,ν2 ,0.05 with
(i) ν1 = 7 and ν2 = 15,
(ii) ν1 = 15 and ν2 = 7,
(iii) ν1 = 24 and ν2 = 19.
7. A random sample of size 100 is taken from a distribution with the expected value of 76 and
the variance of 256. What is the probability that X̄ will be between 75 and 78?
8. If a 1-gallon can of paint covers on the average 513.3 square feet with a standard deviation
of 31.5 square feet, what is the probability that the mean area covered by a sample of 40 of
these 1-gallon cans will be anywhere from 510 to 520 square feet?
9. If the distribution of the weights of all men traveling by air between Dallas and El Paso has
a mean of 163 pounds and a standard deviation of 18 pounds, what is the probability that
the combined gross weight of 36 men traveling on a plane between these two cities is more
than 6000 pounds?
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10. An electrical firm manufactures light bulbs that have a length of life that is approximately
normally distributed, with the expected value equal to 800 hours and a standard deviation of
40 hours. Find the probability that a random sample of 16 bulbs will have an average life of
less than 775 hours.
11. Given the pdf
(
f (x) =
1/4 if x = 0, 1, 2, 3,
0
otherwise,
find the probability that a random sample of size 36, selected with replacement, will yield an
average greater than 1.4 but less than 1.8.
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