Question

Consider a population with 3 observations: 6, 10 and 12. You use simple random sampling without replacement to select a sample of 2 observations. What is the probability that the sample mean is larger than 8?

Answer 1

Answer:

$\dfrac{2}{3}$

Step-by-step explanation:

Given that:

The number of observations is:

6, 10, 12

If we are to use a simple  random sampling without replacement, then we will have:

(6,10)  (6,12)  (10,12)

Here;

the sample size n = 2

The population size N = 3

For (6,10) ; The sample mean = $\dfrac{6+10}{2}$

= $\dfrac{16}{2}$

= 8

For (6,12) ; The sample mean = $\dfrac{6+12}{2}$

= $\dfrac{18}{2}$

= 9

For (10, 12) ; The sample mean = $\dfrac{10+12}{2}$

= $\dfrac{22}{2}$

= 11

The probability distribution of sample mean(x) is:

X           8           9          11

P(X=x)    $\dfrac{1}{3}$          $\dfrac{1}{3}$           $\dfrac{1}{3}$

Thus, the probability that the sample mean  is larger than 8 is:

P(X> 8) = P(X = 9) + P(X + 11)

P(X> 8) = $\dfrac{1}{3}+\dfrac{1}{3}$

P(X > 8) = $\dfrac{1+1}{3}$

P(X> 8) = $\dfrac{2}{3}$