Question

A sample of n=400 observations is drawn from a population with mean μ=1,000 and σ=400. Find the following probabilities:

P
( x ¯ ≤ 1030 )
is:

Answer 1

Probability is the chance that a particular event will occur. Probability ca be estimated using many methods among them being the estimation of probability using the normal distribution curve by the use of the Z-score.

 $\text{Z-score}=\dfrac{X-\mu}{\dfrac{\sigma}{\sqrt{n}}}$

Substituting the given information in our formula above we get that the Z-score is:

$\text{Z-score}=\dfrac{1030-1000}{\dfrac{400}{\sqrt{400}}}$

$\text{Z-score}=1.5$

From the normal standard table the probability that the mean lies between 1030mand 1000 is 0.4332 and the probability that the mean will be less than 1000 is 0.5 therefore, the probability that the mean will be equal to or less than 1030 is

$0.5+0.4332=0.9332$

$P ( x ¯ ≤ 1030 ) =0.9332$