Question

The SAT scores have an average of 1200 with a standard deviation of 60. A sample of 36 scores is selected. a) What is the probability that the sample mean will be larger than 1224

Answer 1

Answer:

the probability that the sample mean will be larger than 1224 is  0.0082

Step-by-step explanation:

Given that:

The SAT scores have an average of 1200

with a  standard deviation of 60

also; a sample of 36 scores is selected

The objective is to determine  the probability that the sample mean will be larger than 1224

Assuming X to be the random variable that represents the SAT score of each student.

This implies that  ;

$S \sim N ( 1200,60)$

the probability that the sample mean will be larger than 1224 will now be:

$P(\overline X > 1224) = P(\dfrac{\overline X - \mu }{\dfrac{\sigma}{\sqrt{n}} }> \dfrac{}{}\dfrac{1224- \mu }{\dfrac{\sigma}{\sqrt{n}} })$

$P(\overline X > 1224) = P(Z > \dfrac{1224- 1200 }{\dfrac{60}{\sqrt{36}} })$

$P(\overline X > 1224) = P(Z > \dfrac{24 }{\dfrac{60}{6} })$

$P(\overline X > 1224) = P(Z > \dfrac{24 }{10} })$

$P(\overline X > 1224) = P(Z > 2.4 })$

$P(\overline X > 1224) =1 - P(Z \leq 2.4 })$

From Excel Table ; Using the formula (=NORMDIST(2.4))

P(\overline X > 1224) = 1 -  0.9918

P(\overline X > 1224) = 0.0082

Hence;  the probability that the sample mean will be larger than 1224 is  0.0082