Transcript ex_sampling_theory - University of Zimbabwe

Econ 110: Sampling Theory and
Statistical Inference In Economics
2nd semester 2016
Richard makoto
Economics department
University of Zimbabwe
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Sampling Theory Exercises
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• A cereal packing machine is calibrated to pack
a mean weight of 500g into carton boxes.
There is some variation in the mean weight
packed per box as measured by a standard
deviation of 10g. Suppose the weight packed
by the machine follows a normal distribution.
What is the probability that a box will weigh
• i) Less than 525g
• ii) More than 480g
• iii) More than 520g but less than 525g
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The mean annual cost of automobile insurance is
$939. Assume that the standard deviation is σ=
$245.
What is the probability that a simple random
sample of automobile insurance policies will
have a sample mean within $25 of the
population mean for each of the following
sample sizes: 30, 50, 100, and 400?
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The average score for male golfers is 95 and the
average score for female golfers is 106. Use these
values as the population means for men and women
and assume that the population standard deviation is
σ=14 strokes for both. A simple random sample of 30
male golfers and another simple random sample of
45 female golfers will be taken.
a. Show the sampling distribution of x-bar for male
golfers.
b. What is the probability that the sample mean is
within 3 strokes of the population mean for the
sample of male golfers?
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To estimate the mean age for a population of 4000
employees, a simple random sample of 40
employees is selected.
a. Would you use the finite population correction factor
in calculating the standard error of the mean?
Explain.
b. If the population standard deviation is σ= 8.2 years,
compute the standard error both with and without
the finite population correction factor. What is the
rationale for ignoring the finite population correction
factor whenever n/N< .05?
c. What is the probability that the sample mean age of
the employees will be within
+/-2 years of the population mean age?
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A simple random sample of size 100 is selected
from a population with p= .40.
• Show the sampling distribution of p-bar.
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The president of Doerman Distributors, Inc.,
believes that 30% of the firm’s orders come
from first-time customers. Arandom sample of
100 orders will be used to estimate the
proportion of first-time customers.
a. Assume that the president is correct and p=
.30. What is the sampling distribution of for
this study?
b. What is the probability that the sample
proportion will be between .20 and .40?
c. What is the probability that the sample
proportion will be between .25 and .35?
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• When production is operating correctly, the
resistence in ohms of components produced
has a normal distribution with std deviation
3.6. Arandom sample of four components was
taken. What is the probability that the sample
variance is bigger than 30.
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• Tom and George are playing in the club golf
tournament.Their scores vary as they play the
course repeatedly.Tom’s score X has the
N(110,10) distribution,and George’s score Y
varies from round to round according to the
N(100,8)
distribution.
If
they
play
independently, what is the probability that
Tom will score lower than George and thus do
better in the tournament?
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zimbabwe
Chap 20-10
Solution
• The difference X −Y between their scores is
Normally distributed, with mean and variance
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zimbabwe
Chap 20-11
• The U.S. Department of Agriculture (USDA) uses
sample surveys to produce important economic
estimates. One pilot study Estimated wheat
prices in July and in September using
independent samples of wheat producers in the
two months. Here are the summary statistics, in
dollars per tonne.
Month
n
x-bar
`s
September 45
$3.61
`$0.19
July
90
$2.95
$0.22
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zimbabwe
Chap 20-12
• Find the probability that the mean of
September is greater than the mean of July.
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Chap 20-13
• Here are some summary statistics from two
independent samples from Normal distributions:
Sample
n
s2
1
2
10
16
3.1
9.3
• You want to test the null hypothesis that the two
population standard deviations are equal at the
5% significance level.
• (a) Calculate the test statistic.
• (b) Find the appropriate value from tables that
You need to perform the significance test.
• (c) What do you conclude?
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Chap 20-14