# Probability and Statistics Questions and Answers – Sampling Distribution of Proportions

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This set of Probability and Statistics Questions and Answers for Campus interviews focuses on “Sampling Distribution of Proportions”.

1. The probability of selecting a sample containing n items from a population with N items without replacement in a Sampling Distribution is?
a) 1/NCn
b) 1/nCN
c) 1/2n
d) 1/2N

Explanation: The number of ways of selecting and samples of size n from a population containing N atoms is NCn. The probability of selecting of each sample is 1/NCn.

2. Find the number of all possible samples from a population containing 18 items from which 6 items are selected at random without replacement.
a) 18564
b) 15864
c) 20264
d) 21564

Explanation: The number of ways of selecting n samples from a population containing n items is NCn. The population is N = 18 and sample size is n = 6. Therefore the number of possible samples are 18C6 = 18564.

3. A pack of cards contains 52 cards. A player selects 4 cards at random without replacement. Find all possible combinations of the cards selected.
a) 207752
b) 270752
c) 270725
d) 207725

Explanation: Considering the experiment to be a sampling distribution where the population contains 52 cards and each sample contains 4 cards. The number of possible samples without replacement are NCn that is 52C4 = 207725 samples.
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4. A population contains N items out of which n items are selected with replacement. Find the probability of the sample being selected.
a) 1/N
b) 1/nN
c) 1/NCn
d) 1/Nn

Explanation: The number of samples containing n items selected from a population of N items is Nn. The probability of selection of each sample is 1/Nn.

5. A box contains 26 pairs of napkins. If 3 pairs of napkins are selected at random with a replacement then the number of possible samples is _______
a) 17675
b) 17566
c) 17576
d) 17556

Explanation: The number of samples formed with n items from a population containing N items is Nn.
Here N = 26 and n = 3.
Hence samples are Nn = 263 = 17576.

6. A sample was formed consisting of 8 students from a total of 56 students for certain task. Find the sampling fraction of the population of students.
a) 1/7
b) 7
c) 49
d) 1/49

Explanation: In a sampling distribution if N is the population size, n is the sample size then number of sampling fractions is n/N. Hence N=56 and n=8 which gives n/N as 8/56 = 1/7.

7. Find the population proportion p for an IPL team having total 30 players with 10 overseas players.
a) 1/2
b) 1/3
c) 2/3
d) 1/4

Explanation: The population proportion p for a population consisting of N items with X specialized items is given as X/N. Now N=30, X=10. Hence p = X/N = 10/30 = 1/3.

8. It is provided that for a sampling distribution E(X)=11 and ϕ=13. Find the bias in the sampling.
a) 2
b) 4
c) 6
d) 3

Explanation: The bias for sampling distribution is given as |E(X) – ϕ|.
|11 – 13| = 2
Hence bias is 2.

9. Find the standard error of population proportion p for sampling with replacement. The population proportion is 0.5 and size of sample is 4.
a) 0.5
b) 0.25
c) 0.225
d) 0.375

Explanation: The standard error of population proportion in sampling distribution with replacement is given as [p*(1-p)/n]1/2. Here p=0.5 and n=4. Hence substituting the values we get
[0.5*(1-0.5)/4]1/2
= 0.25.

10. Find the value of standard error Ẋ in a sampling distribution without replacement. Given that the standard deviation of the population of 100 items is 25.
a) 3
b) 4
c) 2
d) 5

Explanation: Standard error in a sampling distribution with replacement is given by Ẋ = σ/(n)1/2. The standard error with the replacement of items remains the same that is Ẋ = σ/(n)1/2.
Hence n = 100 and σ = 25
Ẋ = σ/(n)1/2
Ẋ = 25/(100)1/2
Ẋ = 2.5
The value of Ẋ = 2.5.

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