Statistics and Probability Answers

Suppose that Q 1 , Q 2 , ∧ and Q 3 are ∧ estimators of ∧ θ. We know
∧
∧
that E ( Q 1 ) = E ( Q 2 ) = θ , E ( Q 3 ) ≠ θ ,V ( Q 1 ) = 12 , V ( Q 2 ) = 10 and
∧
E( Q 3 − θ ) 2 = 6 . Compare these three estimators. Which do you
prefer? Why?

A consumer electronics company is comparing the bright-
ness of two different types of picture tubes for use in its television
sets. Tube type A has mean brightness of 100 and standard devia-
tion of 16, and tube type B has unknown mean brightness, but the
standard deviation is assumed to be identical to that for type A
. A random sample of n = 25 tubes of each type is selected, and
X B − X A is computed. If μ B equals or exceeds μ A , the manufac-
turer would like to adopt type B for use. The observed difference
is x B − x A = 3 . 5 . What decision would you make, and why?

Qn4. Persons suffering from a blood disease are found to have an abnormality in one particular chromosome. However, not all samples of this chromosome are abnormal, and in order to estimate the proportion of affected ones, 5 samples of this chromosome are examined from each of 120 patients, the number of affected one, r, being recorded for each patient. The results are tabulated below;
r 0 1 2 3 4 5
frequency (f) 6 31 42 29 10 2

REQUIRED:
i.) Estimate the proportion 'p' of affected chromosomes and hence
ii.) Compute their variance. (4 Marks)

Qn.3 Consider tossing 4 coins, which are properly balanced so that the probability of a head at each toss is P=1/2. Let r be the number of heads resulting from the 4 tosses.
REQUIRED:
i.) Construct the probability distribution for r. (4 Marks)
ii.) Compute the expected value and variance for r. (2 Marks

Qnb.The number of snakes found in each of 100 sampling quadrats in an area were as follows:
Number of snakes, S 0 1 2 3 4 5 8 15
f = frequency of S 69 18 7 2 1 1 1 1

REQUIRED
Find the mean and variance of the number of snakes per sampling unit.
Determine whether or not the data fit the Poisson distribution. (4 Marks)

Question
a. A radioactive source is emitting, on the average, one particle per minute. If counting continues for several hundred minutes, during which time the particles are emitted randomly, in what proportion of these minutes is to be expected that
i. There will be 2 or more particles emitted? (3 Marks)
ii. There will be 2 or more particles emitted in 2 minutes? (3 Marks)

A research firm conducted a survey to determine the mean amount of money smokers spend on cigarettes during a day. A sample of 100 smokers revealed that the sample mean is N$ 5.24 and sample standard deviation is N$ 2.18 Assume that the sample was drawn from a normal population.
2.1) Find the point estimate of the population mean

The table below shows the heights (in meters) of a random sample of seven cedar trees.



Tree

A

B

C

D

E

F

G

Height

15

13

11

9

7

5

10



Use the data provided to estimate the unknown variance of the entire population of cedar trees with a 99 % degree of confidence.

1. A new pub has opened in town and you’re considering it as a potential new spot for cheap wings and beer. Before going, you chat with a group of 12 friends who have already been. You obtain the following information:
Three of them complained about the quality of the beer.
Four of them complained about the quality of the wings.
Two of them complained about the quality of the beer and wings.
Three of them had no complaints.
Answer the following questions:
a. What is the probability you will enjoy the beer?
b. What is the probability you will enjoy the wings?
c. What is the probability you will dislike either the wings, or the beer, or both?
d. What is the probability you will enjoy both the beer and the wings?
e. Why have twelve of your friends already been here, but none have invited you?

Suppose you just received a shipment of thirteen televisions. Two of the televisions are defective. If two televisions are randomly​ selected, compute the probability that both televisions work. What is the probability at least one of the two televisions does not​ work The probability that at least one of the two televisions does not work?