Statistics and Probability Answers

8b) Suppose from a total of 120 guava trees, 5 clusters of 4 trees each selected and the

yield (in kg) is recorded below:

Cluster Trees

1 2 3 4

1 5 4 2 15

2 11 1 4 7

3 26 10 19 11

4 7 15 12 10

5 2 22 8 6

Estimate the average yield per tree and its standard error.

8. a) A shop manufacturer is trying to determine whether or not to market a new type of

shop. It chooses a random samples of equivalent size in England and France. It asks

each of the people in each sample to try the new soap, and see whether he or she

likes it better than other soaps. The results are as follows:

United States England France

Prefer new soap 81 43 26

Do not prefer it 219 257 274

Test the hypothesis that there are international difference in the proportion of people

who prefer the new soap, using 5% level of significance.

7b) Write three advantage and three disadvantage of using a sampling approach instead

of a census approach for studying a characteristic.

7. a) At a call centre, callers have to wait till an operator is ready to take their call. To

monitor this process, 5 calls were recorded every hour for the 8-hour working day.

The data below shows the waiting time in weeks:

Time Sample number

1 2 3 4 5

9 a.m. 8 9 15 4 11

10 7 10 7 6 8

11 11 12 10 9 10

12 12 8 6 9 12

1 p.m. 11 10 6 14 11

2 7 7 10 4 11

3 10 7 4 10 10

4 8 11 11 11 7

Compute the control limits for x and R -charts. Draw the control charts for x and

,R and comments.

6c) In a television appearance, an MLA from a state asks voters in that state to indicate

whether the MLA should vote against or for a particular piece of legislation. The

MLA office receives 4100 replies. Since voters from western part tends to be

democrats while those from eastern parts tend to e communists, the MLA divides the

replies on the basis of their geographical origin. 68% of the replies from the western

part are against the legislation and 36% of the replies from the eastern part are

against it.

(i) Did the MLA carry out a stratified sampling?

(ii) Did the MLA carry out a stratified random sampling?

(iii) How would you justify the sample design?

(iv) What improvements would you suggest?

6b) The following table gives the hours spent by 50 students in the playground in a week:

Hours 0 2 4 6 8 10

No. of students 10 6 8 6 10 10

Calculate the mean median and standard deviation of the hours required

by students.

6. a) Find the probability that at most 5 defective fuses will be found in a box of 200, if

experience shows that 20% of such fuses are defective.

5b) It is desired to estimate average annual wool yield per sheep for a herd of 150 at a

certain farm house using stratified simple random sampling. Sheeps in the herd are to

be grouped into three strata on the basis of first shaving in days. Optimum method

sample allocation is said to be used for selecting the overall sample of 25 sheeps

from the three strata. Determine approximately optimum strata boundaries using the

information on first shaving given below.

Shaving days No. of Sheep 

)f(

f

 30-70 6 2⋅ 45

70-110 2 1⋅414

110-150 1 1 

150-190 4 2 

190-230 2 1⋅414

230-270 7 2⋅646

270-310 8 2⋅828

310-350 20 4⋅ 472

350-390 10 3⋅162

390-430 25 5 

430-470 40 6⋅325

470-510 25 5

5. a) Consider the case of tossing two dices. Let

: A be the event of getting an odd total

: B the event of getting on the first dice, and

: C the event of getting a total of seven on two dices.

Then

(i) Find (P A ∩C), (P A ∩ )B and (P B∩ C).

Check whether A,B and C are independent or not. Give reasons for your answers.

4b) Consider a population of 11 schools from which a sample of size 3 is to be selected.

List all the possible samples by circular systematic sampling.