Description
STA219: Probability and Statistics for Engineering Assignment 1 with Solution
1. (5 points) Suppose that after 10 years of service, 40% of computers have problems with
motherboards (MB), 30% have problems with hard drives (HD), and 15% have problems with both
MB and HD. What is the probability that a 10-year old computer still has fully functioning MB and
HD?
2. (15 points) Among programmers of a certain firm, 70% know Java, 60% know Python, and 50%
know both languages. If we randomly sample a programmer from the firm, compute the probability
of the following events:
(1) The probability that he/she does not know Python and does not know Java. (5 points)
(2) The probability that he/she knows Java but not Python. (5 points)
(3) The probability that he/she knows Java given that he/she knows Python. (5 points)
3. (10 points) Please derive the permutation and combination under the case of random selecting with
replacement.
4. (15 points) Randomly pick 2𝑘 shoes from 𝑛 pairs of shoes (left and right shoes are distinguishable)
of different sizes (2𝑘 < 𝑛). Calculate the probability of the following events:
(1) Exactly 𝑘 pairs are formed among the 2𝑘 shoes picked. (5 points)
(2) No pair is formed among the 2𝑘 shoes picked. (5 points)
(3) Exactly one pair is formed among the 2𝑘 shoes picked. (5 points)
5. (10 points) Suppose that 4 male-female couples are at a party and that the males and females are
randomly paired for a dance. Compute the probability that at least one couple is paired together.
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6. (10 points) Take two numbers randomly from the interval [0, 1], try to find the probability of the
event that the sum of these two numbers is less than 7/5.
7. (10 points) Continued with Example 1.13 in the slides of Chapter 1 used in class, if the rare disease
occurs in 1 person in 1,000 for those who have a certain symptom, and other known conditions are
the same. If you have the symptom and the test says you have the disease, what is the probability that
you actually have the disease? How is the probability compared to that in Example 1.13?
8. (15 points) A family has two children. The gender of the two children are independent and a child
being a boy (𝐵) or a girl (𝐺) with equal probability. You randomly meet one of the two children with
equal probability and he is a boy, what is the probability that the other child is also a boy? There are
two contradictory opinions:
(1) Since the gender of the two children are independent, the probability that the other child is also
a boy does not depend on whether the child you met is a boy or a girl, i.e., 1/2.
(2) The original sample space consists of 4 equally likely outcomes {𝐺𝐺,𝐵𝐵, 𝐵𝐺, 𝐺𝐵}. Since you
have already met a boy, the sample space becomes {𝐵𝐵, 𝐵𝐺,𝐺𝐵}. Among the remaining three
outcomes, only 𝐵𝐵 suggests that the other child is also a boy, thus the probability is 1/3.
Which of the opinions is correct? Please provide detailed reason/derivation on why the other opinion
is wrong.
9. (10 points) In the system in the figure below, each of the 5 component fails with probability 0.3
independently of other components. Compute the system’s reliability, i.e., the probability that the
system works properly.
STA219: Probability and Statistics for Engineering Assignment 2 with Solution
1. (15 points) Fischer and Spassky play a chess match in which the first player to win a game wins
the match. After 10 successive draws (平局), the match is declared drawn. Each game is won
by Fischer with probability 0.4, is won by Spassky with probability 0.3, and is a draw with
probability 0.3, independent of previous games.
(1) What is the probability that Fischer wins the match? (5 points)
(2) What is the PMF of the duration of the match? (10 points)
2. (15 points) You just rented a large house and the realtor gave you 5 keys, one for each of the 5
doors of the house. Unfortunately, all keys look identical, so to open the front door, you try
them at random. Find the PMF of the number of trials you will need to open the door, under
the following alternative assumptions:
(1) After an unsuccessful trial, you mark the corresponding key, so that you never try it again.
(10 points)
(2) At each trial you are equally likely to choose any key. (5 points)
3. (10 points) Suppose there are 10 electric elements of the same type, among which 2 are
unqualified. You randomly select one from these 10 elements. If it is unqualified, you throw it
away and select one again from the remaining 9 elements; if the second try is still unqualified,
you select another one from the remaining 8 elements. Let 𝑋 be the number of unqualified
elements you select before you get the qualified one. What is the variance of 𝑋?
4. (15 points) Suppose the PDF of a continuous random variable 𝑋 is given by
𝑓(𝑥) = ‘
𝑎𝑥 + 𝑏𝑥!, 0 < 𝑥 < 1,
0, otherwise.
If the expectation of 𝑋 is E(𝑋) = 2/3, please calculate Var(𝑋).
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5. (10 points) Let 𝑋 be a Poisson random variable with parameter 𝜆. Show that the PMF 𝑝”(𝑘)
increases monotonically with 𝑘 up to the point where 𝑘 reaches the largest integer not
exceeding 𝜆, and after that point decreases monotonically with 𝑘.
6. (10 points) Continued with Example 3.8 in the slides of Chapter 2 – Part 1, calculate the
probability that the insurance company loses money in this life insurance, i.e., the profit is less
than $0.
7. (10 points) Jobs are sent to a printer independently at a constant rate of 3 jobs per hour.
(1) What is the expected time between jobs? (5 points)
(2) Suppose that a job just arrived, what is the probability that the next job is sent within 5
minutes? (5 points)
8. (15 points) Continued with Page 41 of Chapter 2 – Part 2, please write down the relationship
between the geometric distribution and the exponential distribution with detailed derivation.
STA219: Probability and Statistics for Engineering Assignment 3 Solution
1. (10 points) Suppose that the average household income in some country is 900 coins, and the
standard deviation is 200 coins. Assuming the normal distribution of incomes:
(1) Compute the proportion of “the middle class”, i.e., whose income is between 600 and 1200
coins. (5 points)
(2) The government of the country decides to issue food stamps to the poorest 3% of households.
Below what income will families receive food stamps? (5 points)
2. (10 points) Let 𝑋~𝑁(𝜇, 𝜎
2
). Suppose that the probability that the quadratic equation 𝑦
2 + 4𝑦 +
𝑋 = 0 has no real roots (i.e., its discriminant is negative) is 0.5, please determine the value of 𝜇.
3. (10 points) A survey shows that the English score (hundred-mark system) of students approximately
follows a normal distribution 𝑁(𝜇, 𝜎
2
) with 𝜇 = 72. If the number of students with more than 96
points accounts for 2.3% of the total students, what is the probability that the score is between 60
and 84 points?
4. (10 points) Suppose that the diameter of a disc follows a uniform distribution on (𝑎, 𝑏), what is the
expected area of this disc?
5. (10 points) Let 𝑍~𝑁(0, 1). Find E(Φ(𝑍)) and Var(Φ(𝑍)), where Φ is the CDF of 𝑍.
6. (10 points) If 𝑋~𝑁(0, 1), please derive the PDF of the following random variables:
(1) 𝑌1 = |𝑋|; (5 points)
(2) 𝑌2 = 2𝑋
2 + 1. (5 points)
7. (15 points) Suppose that random variable 𝑋 follows an exponential distribution with parameter 2.
Show that both 𝑌1 = 𝑒
−2𝑋
and 𝑌2 = 1 − 𝑒
−2𝑋
follow the uniform distribution on (0,1).
PAGE 2/2
8. (15 points) Suppose the joint PDF of a random vector (𝑋, 𝑌) is given by
𝑓(𝑥, 𝑦) = {
𝑘𝑒
−(3𝑥+4𝑦)
, 0 < 𝑥, 𝑦 < ∞
0, otherwise.
(1) Determine the constant 𝑘; (5 points)
(2) Find the joint CDF 𝐹(𝑥, 𝑦) of (𝑋, 𝑌); (5 points)
(3) Compute 𝑃(𝑋 + 𝑌 ≤ 1). (5 points)
9. (10 points) Suppose the joint PDF of a random vector (𝑋, 𝑌) is given by
𝑓(𝑥, 𝑦) = {
𝑒
−𝑦
, 0 < 𝑥 < 𝑦 < ∞
0, otherwise,
determine 𝑓𝑋(𝑥) and 𝑓𝑌(𝑦), i.e., the marginal PDFs of 𝑋 and 𝑌, respectively.
STA219: Probability and Statistics for Engineering Assignment 4 Solutions
.
1. (10 points) Suppose the PDF of random variable Y is
𝑓𝑌
(𝑦) = {
5𝑦
4
, 0 < 𝑦 < 1,
0, otherwise.
Also, the conditional PDF of 𝑋 given 𝑌 = 𝑦 is
𝑓𝑋|𝑌
(𝑥|𝑦) = {
3𝑥
2
𝑦
3
, 0 < 𝑥 < 𝑦 < 1,
0, otherwise.
What is the probability 𝑃(𝑋 > 0.5)?
2. (10 points) Suppose 𝑋 and 𝑌 are independent discrete random variables, and the joint PMF of
(𝑋, 𝑌) is as follows:
X\Y 𝑦1 𝑦2 𝑦3
𝑥1 𝑎 1/9 𝑐
𝑥2 1/9 𝑏 1/3
Please calculate the values of 𝑎, 𝑏, 𝑐.
3. (10 points) Let 𝑋 and 𝑌 be independent Poisson random variables with parameter 𝜆 , i.e.,
𝑋~Poisson(𝜆), 𝑌~Poisson(𝜆). Let 𝑈 = 2𝑋 + 𝑌, 𝑉 = 2𝑋 − 𝑌. Please calculate the correlation
coefficient between 𝑈 and 𝑉.
4. (10 points) Toss a coin 𝑛 times, with 𝑋 and 𝑌 representing the number of heads and tails,
respectively. What is the covariance and correlation coefficient between 𝑋 and 𝑌?
5. (15 points) Suppose the joint PDF of random vector (𝑋,𝑌) is
𝑓(𝑥, 𝑦) = {
1, |𝑦| < 𝑥, 0 < 𝑥 < 1,
0, otherwise.
(1) Please calculate E(𝑋), E(𝑌), Cov(𝑋, 𝑌). (10 points)
(2) Is 𝑋 and 𝑌 independent? (5 points)
PAGE 2/2
6. (10 points) Let 𝑋 and 𝑌 be independent Uniform random variables, i.e., 𝑋~U(0,1), 𝑌~U(0,1).
What is the PDF of 𝑇 = 𝑋 + 𝑌?
7. (10 points) Let 𝑋1
, 𝑋2
and 𝑋3
represent the time (in minutes) necessary to perform three
successive repair tasks at a service facility. They are independent, normal random variables with
expected values 45, 50 and 75, and variances 10, 12 and 14, respectively. What is the probability that
the service facility can finish all three tasks within 3 hours (that is, 180 minutes)?
8. (10 points) There are 40 light bulbs, and the lifespan of each light bulb follows an exponential
distribution with an average lifespan of 25 days. Suppose that we use one light bulb at a time and
replace it immediately with a new bulb once the previous one breaks. Please find the probability that
these bulbs can be used for a total of more than 900 days.
9. (15 points) A large hotel has a total of 500 rooms, and each room has one air conditioner with a
power rating of 2 kW. Suppose the occupancy rate is 80%, which means that each room has an 80%
probability of being occupied, independently of other rooms. How many kW of power are needed to
ensure a 99% probability of having enough power for the air conditioners?
STA219: Probability and Statistics for Engineering
Assignment 5
Note: The assignment can be answered in Chinese or English, either is fine. Please provide
derivation and computation details, not just the final answer. Please submit a PDF file on BB.
1. (10 points) 𝑋 and 𝑌 are independent random variables, and 𝑋~𝐸𝑥𝑝(𝜆), 𝑌~𝐸𝑥𝑝(𝜇). Define
random variable 𝑍 as follows:
𝑍 = .
1, 𝑖𝑓 𝑋 ≤ 𝑌,
0, 𝑖𝑓 𝑋 > 𝑌.
Derive the PMF of Z.
2. (10 points) Suppose 𝑋 and 𝑌 are independent, identically distributed random variables, and
follow geometric distribution, which is 𝑃(𝑋 = 𝑘) = (1 − 𝑝)!”#𝑝, 𝑘 = 1,2, …. Derive the PMF
of 𝑍 = max (𝑋, 𝑌).
3. (10 points) Consider a device equipped with 3 identical electrical components that operate
independently. The operating times of these components follow an exponential distribution
with parameter 𝜆. The device functions normally only when all 3 components are working
properly. Please find the PDF of the operating time 𝑇 of the device.
4. (20 points) Suppose 𝑋 and 𝑌 are bivariate normally distributed with means 𝜇$ = 𝜇% = 0,
variances 𝜎$
& = 𝜎%
& = 1, and correlation coefficient 𝜌.
(1) Please derive the distribution and PDF of 𝑋 − 𝑌. (10 points)
(2) Please calculate the covariance and correlation coefficient of 𝑋 − 𝑌 and 𝑋𝑌. (10 points)
5. (15 points) The PDF of zero mean bivariate normal random vector (𝑋, 𝑌)’ is of the form
𝑓(𝑥, 𝑦) = 1
2𝜋𝜎$𝜎%D1 − 𝜌& 𝑒𝑥𝑝 F− 1
2(1 − 𝜌&)
G
𝑥&
𝜎$
& − 2𝜌
𝑥𝑦
𝜎$𝜎%
+ 𝑦&
𝜎%
&IJ.
(1) Please derive the marginal density function of Y. (5 points)
(2) Prove that the conditional PDF of 𝑋 given 𝑌 = 𝑦 is normal, and identify its conditional
mean and variance. (10 points)
(For both questions, please provide a detailed proof. Do not directly use the conclusion from
the slide.)
PAGE 2/2
6. (20 points) The annual revenues of Company X and Company Y are positively correlated since
the correlation coefficient between the two revenues is 0.65. The annual revenue of Company
X is, on average, 4500 with standard deviation 1500. The annual revenue of Company Y is, on
average, 5500 with standard deviation 2000. Assume that X and Y are bivariate normally
distributed.
(1) Calculate the probability that annual revenue of Company X is less than 6800 given that
the annual revenue of Company Y is 6800. (10 points)
(2) Calculate the probability that the annual revenue of Company X is greater than that of
Company Y given that their total revenue is 12000. (10 points)
7. (15 points) Since both the normal distribution and the Poisson distribution can be used to
approximate binomial distribution, can we use normal distribution to approximate the Poisson
distribution?
(1) First derive the normal approximation to the Poisson distribution 𝑃𝑜𝑖𝑠𝑠𝑜𝑛(𝜆). (5 points)
(2) Use Python to plot the Poisson distribution and its corresponding normal approximation
under different values of 𝜆 (similar to the plots on page 51 of the slide). Explain in which
cases the normal approximation works well and in which cases it does not work well. (10
points)
(Please provide both the plots and the code.)
(既然正态分布和泊松分布都能近似⼆项分布，那正态分布是不是也可以近似泊松分布？
请给出泊松分布𝑃𝑜𝑖𝑠𝑠𝑜𝑛(𝜆)的正态近似，并用 Python 代码画出 𝜆 取不同的值时与其近似
正态分布的图形(与 PPT 上类似的图)，说明什么情况下正态近似比较好，什么情况下不太
好。)
STA219: Probability and Statistics for Engineering Assignment 6 with Solution
Part I: Calculations and derivations by hand
1. (10 points) Bob and Carl have just learned the Law of Large Numbers. It turns out that they
have a different understanding of what the law says.
Bob: The Law of Large Numbers says that in the long run, a fair coin will land heads as often
as it lands tails.
Carl: I don’t think that’s what it says. The Law of Large Numbers says that the fraction of
heads will get closer and closer to 1/2, which is the expected value of each toss.
Bob: Isn’t that the same as what I said?
Carl: No, you said, “The coin will land heads as often as it lands tails,” which implies that the
difference between the number of heads and the number of tails will get smaller as we toss the
coin more and more. I don’t think the number of heads will be close to the number of tails.
Bob: If the fraction of heads is close to 1/2, then the number of heads must be close to the
number of tails. How could it be otherwise?
Who is right: Bob or Carl?
(1) Calculate the variance of
number of heads in 𝑛 tosses − number of tails in 𝑛 tosses
as a function of 𝑛. (5 points)
(2) Considering this calculation, do you agree with Bob that the difference between the
number of heads and the number of tails approaches 0 as the number of tosses increases?
(5 points)
2. (15 points) Let 𝑋!, 𝑋”, … , 𝑋# be a sequence of independent and identically distributed random
variables. Define 𝑌# = (𝑋! + 𝑋” + ⋯ + 𝑋#)/𝑛. Show that the sequence 𝑌!, … , 𝑌# converges
in probability to some limit, and identify the limit, for each of the following cases:
(1) 𝑋$ follow the Poisson distribution with parameter 3. (5 points)
(2) 𝑋$ are uniformly distributed over [−1,3]. (5 points)
(3) 𝑋$ follow the exponential distribution with parameter 5. (5 points)
PAGE 2/3
3. (15 points) Continued with Example 4.5 (3), prove that if 𝑈!~U[0,1] and 𝑈”~U[0,1] are
independent, and let
G
𝑍! = I−2 ln(𝑈!) cos(2𝜋𝑈”)
𝑍” = I−2 ln(𝑈!) sin(2𝜋𝑈”)
,
then 𝑍! and 𝑍” are a pair of independent standard normal random variables.
(Hint: Show that𝑃(𝑍! ≤ 𝑎, 𝑍” ≤ 𝑏) = Φ(𝑎)Φ(𝑏) for all 𝑎 and 𝑏, which requires variable
substitution in a double integral.)
4. (10 points) Explain how to generate numbers from the following distributions based on a
uniform distribution random number generator:
(1) Geometric(p); (5 points)
(2) Standard Cauchy distribution, of which the PDF is 𝑓(𝑥) = 1/[𝜋(1 + 𝑥”)]，𝑥 ∈ (−∞, ∞).
(5 points)
Part II: Implementations by Python
Note: Please provide both the results and the code, and present them in one PDF file.
5. (15 points) Continued with Problem 4, generate 10000 numbers from the following
distributions based on a uniform distribution random number generator in Python, plot the
histogram of the generated numbers, and compare it with the theoretical PMF/PDF of the
following distributions:
(1) Geometric(0.5); (5 points)
(2) Standard Cauchy distribution. (10 points)
6. (15 points) Apply the rejection sampling technique to sample from
𝑓∗(𝑥) = 0.6 exp{−(𝑥 + 5)”/2} + 0.4 exp{−(𝑥 − 1)”/0.5}, 𝑥 ∈ (−∞, ∞).
(1) Choose an appropriate proposal distribution and plot it to show that it covers the target
distribution. (5 points)
(2) Apply the rejection sampling method to generate 500,000 samples, plot the histogram of
the generated samples, and compare it with the theoretical PDF 𝑓(𝑥) (normalized 𝑓∗(𝑥)).
(5 points)
(3) What is the acceptance proportion? (5 points)
PAGE 3/3
7. (20 points) A forest consists of 1,000 trees forming a perfect 20 × 50 rectangle as in the
figure below. The northwestern (top-left) corner tree catches fire. Wind blows from the
northwest, therefore trees can only catch fire from its buring left and above neighbors (⽕只
会向东或向南蔓延). Assume the probability that any tree catches fire from its burning left
neighbor is 0.8, and the probabilities to catch fire from trees immediately to the above is 0.3.
(1) Conduct a Monte Carlo study to estimate the probability that more than 30% of the forest
will eventually be burning. With probability 0.95, your answer should differ from the true
value by no more than 0.005. (10 points)
(You should first determine the value of 𝑛 that satisfies the requirement, and then
conduct a Monte Carlo study to estimate the probability.)
(2) Based on the same study, predict the total number of affected trees 𝑋. (5 points)
(3) What is the corresponding standard deviation of 𝑋. (5 points)
STA219: Probability and Statistics for Engineering Assignment 7 Solution
1. (10 points) A network provider investigates the load of its network. The number of concurrent
users is recorded at ten locations (thousands of people),
17.2 22.1 18.5 17.2 18.6 14.8 21.7 15.8 16.3 22.8
(1) Compute the sample mean, sample variance, and sample standard deviation of the number
of concurrent users. (5 points)
(2) Compute the sample lower and upper quartile, and sample interquartile range. (5 points)
2. (15 points) Let 𝑋!, 𝑋”, 𝑋# be a simple random sample from the population 𝑋~U(0, 𝜃).
(1) Show that 𝜃*
! = $
#
𝑋(#) and 𝜃*
” = 4𝑋(!) are both unbiased estimators of 𝜃. (10 points)
(2) Which of these two estimators is more efficient? (5 points)
3. (10 points) The average white blood cell count per liter of blood in normal adult males is
7.3 × 10′, with a standard deviation of 0.7 × 10′. Using Chebyshev’s inequality, estimate the
lower bound for the probability that the white blood cell count per liter of blood is between
5.2 × 10′ and 9.4 × 10′.
4. (10 points) Let 𝑋!, … , 𝑋( be a simple random sample from the population 𝑋, and E(𝑋) =
𝜇, Var(𝑋) < ∞. Prove that
𝜇̂= 2
𝑛(𝑛 + 1)
@𝑘𝑋)
(
)*!
is a consistent estimator of 𝜇.
(Hint 1: use the conclusion that asymptotic unbiasedness + vanishing variance ⟹
consistency; Hint 2: ∑ 𝑖 ( ”
+*! = 𝑛(𝑛 + 1)(2𝑛 + 1)/6.)
5. (15 points) Estimate the unknown parameter 𝜃 from a sample
3, 3, 3, 3, 3, 7, 7, 7
drawn from a population 𝑋 with the probability mass function
PAGE 2/2
G
P(𝑋 = 3) = 𝜃;
P(𝑋 = 7) = 1 − 𝜃.
(1) Derive the moment estimator 𝜃*
! of 𝜃, and calculate its estimated value based on the
sample; (5 points)
(2) Calculate the expectation and variance of 𝜃*
!. Is 𝜃*
! an unbiased estimator? (5 points)
(3) Calculate the maximum likelihood estimate of 𝜃 based on the sample. (5 points)
6. (15 points) A sample (𝑋!, . . . , 𝑋!,) is drawn from a population with a PDF
𝑓(𝑥; 𝜃) = 1
𝜃 𝑒-.
/, 0 < 𝑥 < ∞.
The sum of all 10 sample observed values equals 150.
(1) Derive the moment estimator 𝜃*
! of 𝜃, and calculate its estimated value based on the
sample. (5 points)
(2) Derive the standard error of the moment estimator 𝜃*
!. (5 points)
(3) Derive the maximum likelihood estimator 𝜃*
” of 𝜃, and calculate its estimated value
based on the sample. (5 points)
7. (15 points) Installation of a certain hardware takes random time with a standard deviation of
5 minutes.
(1) A computer technician installs this hardware on 64 different computers, with the average
installation time of 42 minutes. Compute a 95% confidence interval for the population
mean installation time. (10 points)
(2) Suppose that the installation time follows normal distribution, and population mean
installation time is 40 minutes. A technician installs the hardware on your PC. What is
the probability that the installation time will be within the interval computed in (1)? (5
points)
8. (10 points) Assuming that the height of a randomly selected woman aged 18 to 25 follows a
normal distribution. We collected data from two regions, A and B. In region A, 40 women were
sampled with a mean height of 1.64m and a standard deviation of 0.2m. In region B, 50 women
were sampled with a mean height of 1.62m and a standard deviation of 0.4m. Estimate the
difference in mean height between women from these two regions with 90% confidence.
STA219: Probability and Statistics for Engineering Assignment 8 with Solution
1. (20 points) Let 𝑋!, … , 𝑋” be a simple random sample from the population 𝑋~N(𝜇, 1).
Consider the testing problem 𝐻#: 𝜇 = 2 ⟷ 𝐻!: 𝜇 = 3. We choose the rejection region to be
{𝑿: 𝑋3 > 2.6}.
(1) When the sample size 𝑛 = 20, calculate the Type I and Type II error rates. (10 points)
(2) In order to control the Type II error rate below 1%, what is the minimum sample size 𝑛?
(10 points)
2. (15 points) A principal of an elementary school saw a report in the newspaper stating, “Middle
school students in this city watch an average of 8 hours of television per week.” She believes
that the students at her school spend significantly less time on watching TV than the
newspaper stated. To investigate, she randomly surveyed 100 students at her school and
found that the average time spent watching TV per week was 𝑥̅= 6.5 hours, with a sample
standard deviation of 𝑠 = 2 hours. The question is whether it can be concluded that the
principal’s belief is correct.
(1) Construct a rejection region at significance level of 𝛼 = 0.05, and based on the sample
observed value, decide whether to reject the statement in the newspaper. (10 points)
(2) Alternatively, compute the 𝑝 -value of the observed value of the test statistics. At
significance level of 𝛼 = 0.05, is the conclusion the same as in (1)? (5 points)
3. (10 points) (Continued with Problem 3.) Suppose the principal is interested in testing whether
the students at her school spend significantly less time on watching TV than the newspaper
stated (i.e., less than 8 hours per week). She plans to randomly select 𝑛 students and perform
a significance test at the 5% significance level. If the true mean time of her students watching
TV is 7.5 hours and the true standard deviation is 2 hours, what is the minimum sample size
𝑛 necessary to achieve at least 0.90 power in this significance test?
4. (30 points) A news agency publishes results of a recent poll. It reports that a certain candidate
has a 10% stronger support in town A than in town B because 45% of the poll participants in
town A and 35% of the poll participants in twon B supported the candidate. Notice that 900
randomly selected registered voters participated in the poll in each town. Let 𝑝$ and 𝑝% be
the support rates of the candidates in town A and town B.
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(1) At significance level 𝛼 = 0.02, test whether there is a significant difference between the
support rates of the candidate in town A and town B. (5 points)
(2) Calculate the 98% confidence intervals for each of 𝑝$, 𝑝%, and 𝑝$ − 𝑝%. (10 points)
(3) Can we make a decision about whether the the support rates of the candidate in town A
and town B are significantly different or not at level 𝛼 = 0.02 based on whether the 98%
confidence intervals of 𝑝$ and 𝑝% overlap or not? State your explanation. (10 points)
(4) Can we make a decision about whether the the support rates of the candidate in town A
and town B are significantly different or not at level 𝛼 = 0.02 based on whether the 98%
confidence intervals of 𝑝$ − 𝑝% contain 0 or not? State your explanation. (5 points)
5. (25 points) Suppose that 𝑋!, … , 𝑋” is a simple random sample from population 𝑋~N(𝜇!, 4&),
𝑌!, … , 𝑌’ is a simple random sample from population 𝑌~N(𝜇&, 4&). The two samples are
independent. 𝜇!, 𝜇& are unknown population means, and we would like to test 𝐻#: 𝜇! =
𝜇& ⟷ 𝐻!: 𝜇! > 𝜇& at significance level 𝛼 = 0.05.
(1) Choose a proper test statistic 𝑇 and determine its distribution under 𝐻#, then write down
the rejection region based on the test statistic. (5 points)
(2) Suppose that the underlying true mean difference between the two populations is 𝜇! −
𝜇& = 2, calculate the Type II error rate of the test in (1) if 𝑛 = 10 and 𝑚 = 11. (10 points)
(3) Continued with (2), compute the minimum total sample size 𝑁 = 𝑛 + 𝑚 needed to achieve
at least 0.9 statistical power in the test. (10 points)
