Question

worldwide organization of academics claims that the mean IQ score of its members is 112, with a standard deviation of 16. A randomly selected group of 35 members of this organization is tested, and the results reveal that the mean IQ score in this sample is 114.6. If the organization's claim is correct, what is the probability of having a sample mean of 114.6 or less for a random sample of this size

Answer 1

Probability of having a sample mean of 114.6 or less is 0.8315.

Explanation:

We are given that worldwide organization of academics claims that the mean IQ score of its members is 112, with a standard deviation of 16.

A randomly selected group of 35 members of this organization is tested, and the results reveal that the mean IQ score in this sample is 114.6.

Let
`\bar X` = sample mean

The z score probability distribution for sample mean is given by;

Z =
`(\bar X-\mu)/((\sigma)/(√(n) ) )` ~ N(0,1)

where,
`\mu` = population mean IQ score of its members = 112

`\sigma` = standard deviation = 16

n = sample of members = 35

Now, probability of having a sample mean of 114.6 or less is given by = P(
`\bar X`
`\leq` 114.6)

P(
`\bar X`
`\leq` 114.6) = P(
`(\bar X-\mu)/((\sigma)/(√(n) ) )`
`\leq`
`(114.6-112)/((16)/(√(35) ) )` ) = P(Z
`\leq` 0.96)

= 0.8315

Hence, the required probability is 0.8315.