Name:
Score:
Math340 – Midterm # III (A)
1. Suppose that Var(X1 ) = 2, Var(X2 ) = 4, and Cov(X1 , X2 ) = −1. Find
(a) Var(2X1 − 3X2 ).
(b) Cov(2X1 − 3X2 , −X1 + X2 − 2).
2. Let X1 and X2 be a random sample of size n = 2 from the exponential distribution with pdf
f (x) = 2e−2x , 0 < x < ∞.
(a) Find P (0.5 < X1 < 1.0, 0.7 < X2 < 1.2).
(b) Find E (X1 (X2 − 0.5)2 ).
Name:
Score:
Math340 – Midterm # III (A)
3. Let X1 , X2 , and X3 be independent Poisson random variables with respective means λ1 = 1
λ2 = 2 and λ3 = 3. Use the mgf technique to determine the distribution of X1 + X2 + X3
and use the result to find P(X1 + X2 + X3 = 4).
4. Let X̄ be the mean of a random sample of size 81 from an exponential distribution with
mean 3. Approximate P(2.5 ≤ X̄ ≤ 4).
Page 2
Name:
Score:
Math340 – Midterm # III (A)
5. A leakage test was conducted to determine the effectiveness of a seal designed to keep the
inside of a plug airtight. An air needle was inserted into the plug, which was then placed
underwater. Next, the pressure was increased until leakage was observed. The magnitude of
this pressure in psi was recorded for 10 trials:
3.1 3.5 3.3 3.7 4.5 4.2 2.8 3.9 3.5 3.3
(a) Find the sample mean x̄, and the sample median m̃, and the sample standard deviation
s for these 10 measurements.
(b) Construct the 95% confidence interval of the mean of magnitudes of all pressure.
Page 3
Name:
Score:
Math340 – Midterm # III (A)
6. A county’s pastureland is divided into a grid consisting of many, many square yard cells. An
entomologist inspects the ground for egg masses of a harmful insect at 80 randomly selected
cells. She finds egg masses (infestations) in 14 cells.
(a) Find an approximate 95% confidence interval for the proportion all cells in the pastureland that are infested.
(b) If she wants the ± piece of her confidence interval to be 0.025 when she calculates it,
how many cells will she have to inspect? Use 18% as an initial guess for the proportion
of infested cells
Page 4
Name:
Score:
Math340 – Midterm # III (B)
1. Suppose that Var(X1 ) = 4, Var(X2 ) = 2, and Cov(X1 , X2 ) = 1. Find
(a) Var(2X1 − 3X2 + 1).
(b) Cov(2X1 − 3X2 , −X1 + X2 ).
2. Let X1 and X2 be independent Poisson rvs with respective means λ1 = 1 and λ2 = 2.
(a) Find P (0 ≤ X1 ≤ 1, 0 < X2 < 3).
(b) Find E (X1 (X2 − 1)2 ).
Name:
Score:
Math340 – Midterm # III (B)
3. Let X1 , X2 , and X3 be a random sample of size n = 3 from the exponential distribution
with pdf f (x) = 2e−2x , 0 < x < ∞. Use the mgf technique to determine the distribution of
X1 + X2 + X3 and use the result to find P(X1 + X2 + X3 ≤ 1).
4. A candy maker produces mints that have labels weight of 22 grams. Assume that the
distribution of the weights of these mints has a mean 22 and σ62 = 1. A sample of size 36
is drawn, approximate P(21.9 ≤ X̄ ≤ 22.1)
Page 2
(3)
Name:
Score:
Math340 – Midterm # III (B)
5. A leakage test was conducted to determine the effectiveness of a seal designed to keep the
inside of a plug airtight. An air needle was inserted into the plug, which was then placed
underwater. Next, the pressure was increased until leakage was observed. The magnitude of
this pressure in psi was recorded for 10 trials:
3.1 3.5 3.3 3.7 4.5 4.2 2.8 3.9 3.5 3.3
(a) Find the sample mean x̄ and the sample standard deviation s for these 10 measurements.
(b) Construct the 95% confidence interval of the mean of magnitudes of all pressure.
Page 3
Name:
Score:
Math340 – Midterm # III (B)
6. A county’s pastureland is divided into a grid consisting of many, many square yard cells. An
entomologist inspects the ground for egg masses of a harmful insect at 100 randomly selected
cells. She finds egg masses (infestations) in 20 cells.
(a) Find an approximate 95% confidence interval for the proportion all cells in the pastureland that are infested.
(b) If she wants the ± piece of her confidence interval to be 0.025 when she calculates it,
how many cells will she have to inspect? Use 20% as an initial guess for the proportion
of infested cells
Page 4
MATH 340-09 EXAM 3
KIM, SUNGJIN
This is a 70-minute closed-book exam. The textbook, lecture notes, and all personal scratch papers are
not allowed during the test. Discussing with other students or copying other students’ work are strictly
prohibited. Those cases will be reported to the dean’s office. To receive the full credit, you must
explain each step thoroughly. You are not allowed to use your own scratch paper. Include justifications
(52)
for all your answers. For example, if you have the answer 29 and you obtained the answer from 10
, then
(2)
5
(2)
write the intermediate step 10
before writing the answer 92 . You are free to use anything learned in class,
(2)
but make sure you write which one you have used.
Problem 1. Let X equal the lifetime of a smartphone. Assume that µ = 10 (in years) and σ = 3. Let X
be the sample mean of a random sample of n = 30 smartphones. Find the following.
(a) Find E[X]. (4)
(b) Find Var(X). (4)
(c) Approximate the probability P (8 < X < 12). (8)
Problem 2. Let X1 , . . . , X40 be a random sample of size 40 from the distribution with pdf f (x) = 3x2 , 0 <
x < 1. Approximate the probability that at least 2 and at most 8 of these random variables have values
less than 1/2. Apply half-unit correction. Hint: Let the i-th trial be a success if Xi < 1/2, i = 1, 2, . . . , 40.
Then count the number of successes. (18)
Problem 3. The weights (in pounds) of a ”30-pound” dumbbell manufactured by a company are as
follows:
29.12, 28.50, 32.00, 30.90, 31.80, 25.20, 39.60.
(a) Find the five-number summary of the data (the three quartiles, the minimum, and the maximum) and
draw a box-plot. (10)
(b) Find suspected outliers and outliers if they exist. (8)
Problem 4. The following is the sales report of two products that a company A produces. Fill in the
table and find a least squares regression line for this data set. (16)
X Y
XY X 2
1 3.1
2 6.1
3 8.9
4 11.8
Averages
Problem 5. Assuming that the five observations 3.2, 3.5, 2.7, 1.8, and 2.9 are from a normal distribution
N (µ, σ 2 ). Find a 95% confidence interval for µ. (16)
Problem 6. Let p the proportion of icecream cones whose length is shorter than 10 centimiters. Out
of 370 icecream cones, 50 of them are shorter than 10 centimeters. Find an approximate 90% confidence
interval for p. (16)
Date: Dec 2, 2024.
1