Statistics and Probability Answers

4. The mean hemoglobin level in the blood for a certain large group of individuals is 21.0 grams per milliliter (g/ml). The standard deviation is 2 g/ml. If a sample of 25 individuals is selected, find the probability that the mean will be greater than 21.3 g/ml. Assume that the variable is normally distributed.

 Television advertisers value the well-known Radio and TV awards scheme RTP as a measure of a TV/Radio show’s popularity among viewers. The RTP rating of a certain TV program is an estimate of the proportion of viewers, expressed as a percentage, who watch the program on a given day. In one survey, the RTP ratings team found that 101 out of 165 sampled families watched “The KSM Show” on the night of its premiere (the first time it is shown on television). In a separate survey several years later, the RTP ratings team found that 97 out of 180 sampled families watched “Music Music on TV3” on its premiere. (a) Estimate the true proportion p1 of all TV-viewing families who watched the premiere of “The KSM Show” and obtain a 95% confidence interval for p1. (b) Briefly explain what this 95% confidence interval means. (c) If the RTP team wanted to guarantee that all proportions are estimated to within ±0.05 with 95% confidence, how large should their samples be?

The class gained a mean of μ = 24 and a standard deviation of a = 8.48 and X= 6

a. What percentage of scores are lower than your score? Show Steps 1 to 4 completely.

Given population of 5000 scores with mean=86 and standard devation=10. How many scores are between 76 and 86

a population consist of the values(1,4,3,2) consider samples of size 2 that can be drawn from this population

The top-selling Amar tire is rated 70,000 KMs, which means nothing. In fact, the distance the tires can run until they wear out is a normally distributed random variable with a mean of 82,000 KMs and a standard deviation of 6,400 KMs. What is the probability that a tire wears out before 70,000 KMs? What is the probability that a tire lasts more than 100,000 KMs? Note: You may use Z-table for this

A10. If X and Y are independent binomial random variables with identical parameters n

and p, show analytically that the conditional probability of X, given that X + Y = m

is the hypergeometric distribution.

Suppose that a random variable X has a Poisson distribution with parameter λ. The

parameter λ itself is a random variable with the exponential distribution with mean 1

c ,

where c is a constant. Show that

P(X = k) =

c

(c + 1)k+1

) Prove that for any discrete bivariate random variable (X, N) for which the first

moments of X and N exists,

E(X) = E [E (X|N)]

(b) The number N of customers entering the University of Ghana book-shop each day

is a random variable. Suppose that each customer has, independently of other

customers, a probability θ of buying at least one book. Let X denote the number

of customers that buy at least one book each day.

Describe without proof the distribution of X conditional on N = n. Hence use the

results in (a) to evaluate the expectation of X if N has the distribution.

i. P(N = k) = M

k
θk

(1 − θ)M−k

, k = 0, 1, · · · , M

ii. P(N = k) = θ(1 − θ)k

, k = 0, 1, 2, · · · ,

iii. P(N = k) =

e−θθk

k!

, k = 0, 1, 2, · · ·

iv. P(N = k) = θ(1 − θ)k−1

, k = 1, 2, · · ·

Find the probability distribution of X if N has the distribution in (b) i-iv.

Let F and G be two sigma-fields on Ω. Prove that F ∩ G is also a sigma-field on Ω.

Show by example that F ∪ G may fail to be sigma-field if Ω = {1, 2, 3}.