Question

For a population of five individuals, bike ownership is as follows:
(A) = 2; (B) = 1; (C) = 3; (D) = 4; (E) = 2
Determine the probability distribution for the discrete random variable, x = # bikes:
(1) Calculate the population mean.
(2) Calculate the population standard deviation.
(3) For a sample size n=2, determine the mean number of bikes for the two person pair.
(4) How many two person outcomes lead to a mean of 1.5 (note: for consistency, count (A,B) and (B,A) as two separate outcomes)?
(5) What is the P(x̅) = 1.5?
(6) What is the mean of this sampling distribution (n=2)?
(7) What is the standard deviation of this sampling distribution (n=2)?

Answer 1

Probability distribution is,

P(X = 1) = 1/5

P(X = 2) = 2/5

P(X = 3) = 1/5

P(X = 4) = 1/5

(1)

Mean = E(X) = 1 * 1/5 + 2 * 2/5 + 3 * 1/5 + 4 * 1/5 = 2.4

(2)

Standard deviation = = 1.02

(3)

The possible samples of size, n = 2

(A, B) (B, A) (B, E) (E, B) with mean = (2 + 1)/2 = 1.5

(A, C) (C, A) (C, E) (E, C) with mean = (2 + 3)/2 = 2.5

(A, D) (D, A) (D, E) (E, D) with mean = (2 + 4)/2 = 3.5

(A, E) (E, A)  with mean = (2 + 2)/2 = 2

(B, C) (C, B)  with mean = (1 + 3)/2 = 2

(B, D) (D, B)  with mean = (1 + 4)/2 = 2.5

(C, D) (D, C)  with mean = (3 + 4)/2 = 3.5

There are total 20 samples.

(4)

Number of two person outcomes lead to a mean of 1.5 = 4

(5)

P(x̅ = 1.5) = 4/20 = 0.2

(6)

The probability distribution of sample mean is,

P(x̅ = 1.5) = 0.2

P(x̅ = 2) = 4/20 = 0.2

P(x̅ = 2.5) = 6/20 = 0.3

P(x̅ = 3.5) = 6/20 = 0.3

(7)

Mean of this sampling distribution, E(x̅) = 1.5 * 0.2 + 2 * 0.2 + 2.5 * 0.3 + 3.5 * 0.3 = 2.5

(8)

standard deviation of this sampling distribution = = 0.74