Description

 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
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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.