Midterm Exam Practice Problems
1. Let X1, X2, X3 be a random sample from an exponential distribution with mean θ, Let
θˆ1 = X1, θˆ2 = (X1 + X2)/2 and θˆ3 = (X1 + 2X2)/3. Show that these estimators of θ
are all unbiased, and determine the relative efficiencies between them.
2. Let x1 = 7.13, x2 = 5.26, x3 = 9.93, x4 = 6.62, x5 = 7.52 be five numbers from the
gamma distribution(α, β). Use the method of moments to find estimates for α and β.
3. Let X1 = x1, X2 = x2, . . . , Xn = xn denote a random sample from a distribution with
pdf f(x) = θ2
θ
xθ+1
, x ≥ 2, θ > 1. Use the method of moments to estimate θ.
4. Let X1 = 0.4, X2 = 0.5, X3 = 0.25, X4 = 0.9, X5 = 0.92 be a random sample from a
distribution with pdf f(x) = θxθ−1, 0 ≤ x ≤ 1, θ > 0.
(a) Use the method of moments to obtain an estimator of θ, and then compute the
estimate for this data.
(b) Find a sufficient statistic for θ.
(c) Obtain the maximum likelihood estimator of θ, and then compute the estimate
for the given data.
(d) Find a
5. Suppose that a random sample with X1 = 5, X2 = 9, X3 = 9, X4 = 10 is drawn from a
distribution with pdf f(x) = θ
2
√
x
e−θ
√
x, where x > 0. Use the maximum likelihood to
find an estimate for θ.
6. Suppose that the heights of 2-year old girls are normally distributed with a mean of
30 in. and a standard deviation of 6 in.. Let µ denote the true mean heights of these
girls. A researcher wishes to test
H0 : µ = 30 versus HA : µ < 30.
She plans to obtain a random sample of 30 girls and measure their heights, using
α = 0.05 for a significance level. What is the probability of a type II error when
µA = 28 and when µA = 26?
7. A pharmaceutical company is testing a new drug that it hopes will lower cholesterol
levels in patients. They conduct a study in which they test H0: the drug is not
effective in lowering cholesterol levels versus HA: the drug is effective in lowering
cholesterol levels. Describe the Type I and Type II errors in this case and the practical
consequences of making these errors.
8. Suppose you are interested in the length of a certain species of snake in Lyman Lake.
Assume the lengths (in centimeters) are normally distributed with unknown mean µ
but known standard deviation σ = 4. You decide to testH0 : µ = 25 versusHA : µ > 25
at α = 0.05 with a sample size of 30. What is the probability of a type II error for
µ = 27?
9. Your favorite brand of cereal comes in boxes of 570 g. You suspect that the company
is under-filling the boxes. If the true average weight is 561 g (or less), you will contact
the Better Business Bureau. If you decide to test your hypothesis at a 0.05 significance
level, how many boxes of cereal would you need to sample if you want an 80% chance
of detecting a mean of 561 g? Assume for the null hypothesis that cereal weights are
distributed normally with mean µ = 570 and standard deviation σ = 14g.
10. A survey published in the American Journal of Sports Medicine2 reported the num-
ber of meters (m) per week swum by two groups of swimmers—those who competed
exclusively in breaststroke and those who competed in the individual medley (which
includes breaststroke). The number of meters per week practicing the breaststroke
was recorded for each swimmer, and the summary statistics are given below. Is there
sufficient evidence to indicate that the average number of meters per week spent prac-
ticing breaststroke is greater for exclusive breaststrokers than it is for those swimming
individual medley?
Breaststroke Medley
Sample size 130 80
Sample mean 9017 5853
Sample standard deviation 7162 1961
Population mean µ1 µ2
11. The Rockwell hardness index for steel is determined by pressing a diamond point into
the steel and measuring the depth of penetration. For 50 specimens of an alloy of steel,
the Rockwell hardness index averaged 62 with standard deviation 8. The manufacturer
claims that this alloy has an average hardness index of at least 64. Is there sufficient
evidence to refute the manufacturer’s claim at the 1% significance level?
12. The state of California is working very hard to ensure that all elementary age students
whose native language is not English become proficient in English by the sixth grade.
Their progress is monitored each year using the California English Language Devel-
opment test. The results for two school districts in southern California for the 2003
school year are given in the accompanying table. Do the data indicate a significant
difference in the 2003 proportions of students who are fluent in English for the two
districts? Use α = .01.
District Riverside Palm Springs
No. of Students tested 6124 5512
Percentage fluent 40 37
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