Question

A population of scores forms a normal distribution with a mean of μ = 71 and a standard deviation of σ = 11.

(a) What proportion of the scores in the population have values less than X = 69? (Round your answer to four decimal places.)

(b) If samples of size n = 8 are selected from the population, what proportion of the samples will have means less than M = 69? (Round your answer to four decimal places.)

(c) If samples of size n = 31 are selected from the population, what proportion of the samples will have means less than M = 69? (Round your answer to four decimal places.)

Answer 1

Let the random variable X denote the scores.Given that X~ N(71,11)

(a) Standardizing X,  

Using standard normal tables,

We get,

P(X<69) = P(Z<-0.18) = 0.4286 = 42.86%

Proportion of the scores in the population that would have values less than X = 69 is 0.4286

(b) If n = 8,

  

0.3050 = 30.5%

Proportion of the samples that would have means less than M = 69 is 0.3050

(c) For n = 31,

$=P(Z<-1.01)$

  

%

PlM < 69 )= P(Z <-1.01) 0.1562= 15.62

Proportion of the samples that would have means less than M = 69 is 0.1562