Problem 1 [34 points]: Let X ~ N(50.0, 25.0) be the weight (in kg) of an adult female in a

city.

(a) [4 points] If an adult female is selected at random. What is the probability that she is

between 49.75kg and 50.25kg?

(b) [4 points] What is the 33rd percentile of weight from adult female in the city?

(c) [5 points] What is the IQR of weight from adult female in the city?

(d) [3 points] Suppose 625 adult females are selected at random. Let Y be the sample mean

random variable of their weights. What is the distribution of Y, and the corresponding

parameter value(s)?

(e) [4 points] What is the probability that the sample mean of 625 females is between

49.75kg and 50.25kg?

(f) [7 points] What is the probability that the sample mean of 1600 females is between

49.75kg and 50.25kg?

(g) [7 points] How's your answer for part (a), (e) and (f) going to change, if X does not follow

Normal distribution, but instead an unknown continuous probability distribution with

mean 50.0 and s.d. 5.0? Explain.

Problem 2 [27 points]: Consider a coin which is biased towards head, with p=Pr(Head)=0.41.

Suppose we flip a coin 127 times, and let X be the total number of heads being observed.

(a) [4 points] Based on the BINOMDIST function in EXCEL mentioned on p.39 of Ch3,

compute the probability of obtaining more than 46 heads but less than 62 heads.

(b) [5 points] Show that X can be well-approximated by a Normal distribution Y. What are

the parameter values of Y?

(c) [10 points] Based on Normal approximation to Binomial distribution, estimate the

probability of obtaining more than 46 heads but less than 62 heads.

(d) [8 points] Consider a new coin with unknown p = Pr(head). The coin was flipped 500

times, of which 260 times were tails. Obtain a point estimate of p, and construct a 95%

Confidence Interval for p

Problem 3 [20 points]: Consider a continuous random variable X with pdf f(x) defined on

-2 x 6 as the chart below:

(a) [4 points] Show that f(1) = 0.125.

(b) [5 points] Compute Pr(0 X 5).

(c) [5 points] Compute Pr(X 1.5).

(d) [6 points] Suppose F(a) = Pr(X a) = 0.9. What is the value of a?

Problem 4 [19 points]: Consider a fast food chain store which sells fried chicken. A sample

of 12 chicken breast pieces was selected are random, and their weights (in grams) are

recorded as follow:

204

203

188 188 195 202

204 195 180 198

175 208

(

and

).

Let Xi be the weight of the ith chicken breast piece. Assume that each Xi follows normal

distribution with mean and variance 2 .

(a) [5 points] What distribution (and its parameters) does

follow?

(b) [9 points] Construct a 95% confidence interval for .

(c) [5 points] Construct a 95% confidence interval for 2.