Question

Suppose X a population has a distribution that is skewed to the right and µ =400 and σ =24. If a random sample of size 144 is drawn from the population:

   a) What is the name of theorem that allows us to solve this problem? ___________

   b) What is the reason that we use that theorem in this case? ___________

   c) Compute P(399 <  < 403) _________

In part c,   is the sample mean.

PLEASE SHOW ALL YOUR WORK

Answer 1

Solution :

Given that,

mean = $\mu$ = 400

standard deviation = $\sigma$ = 24

n = 144

a) Central Limit Theorem

b) because sample size is greater than or equal 30

$\mu$$\bar x$ = $\mu$ = 400

$\sigma$$\bar x$ = $\sigma$ / $\sqrt$ n = 24/ $\sqrt$ 144 = 2

c) P(399 < $\bar x$ < 403 )  

= P[(399 - 400) / 2 < ($\bar x$ - $\mu$ $\bar x$) / $\sigma$ $\bar x$ < (403 - 400) / 2 )]

= P(-0.50 < Z < 1.50)

= P(Z < 1.50) - P(Z < -0.50)

Using z table,  

= 0.9332 - 0.3085

= 0.6247