Question

A population of values has a distribution with μ=6.6μ=6.6 and σ=8.2σ=8.2. You intend to draw a random sample of size n=112n=112.

According to the Central Limit Theorem:

(a) What is the mean of the distribution of sample means?
μ¯x=μx¯=

(b) What is the standard deviation of the distribution of sample means?
(Report answer accurate to 2 decimal places.)
σ¯x=σx¯=

(c) In a random sample of n=112, what is the probability that its random sample mean is more than 6? Round to three decimal places.



(d) In a random sample of n=112, what is the probability that its random sample mean is less than 7.8? Give your answer to three decimal places.

Answer 1

solution:

Given data

   = 6.6 and  σ = 8.2

When you draw a random sample of size (n) = 112

Then

a)  x = = 6.6

b) Sample standard deviation (x) = /

= 8.2 /

= 0.77

  x = 0.77

c) Probability that random sample mean is more than 6 =

=

= P(Z>-0.7744)

= P(Z<0.7744)

= 0.7807 [using standard normal distribution table ]

Probability that random sample mean is more than 6 = 0.781  

d)  Probability that random sample mean is less than 7.8 =

=

= P(Z<1.5487)

= 0.9393 [using standard normal distribution table ]

Probability that random sample mean is less than 7.8 = 0.939