A worldwide organization of academics claims that the mean 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 117.2. If the organization's claim is correct, what is the probability of having a sample mean of 117.2 or less for a random sample of this size? Carry your immediate computations to at least four decimal places. Round your answer to at least four decimal places. Round your answer to at least three decimal places.

0.973

Step-by-step explanation:

This is a normal distribution problem

μ = mean of population = 112

σ = standard deviation of population = 16

But for this calculation, we require the mean and standard deviation of the sample

μₓ = μ = 112

σₓ = σ/√n = 16/√35 = 2.70 (n = sample size)

We then standardize the IQ score of 117.2

The standardized score for any value is the value minus the mean then divided by the standard deviation.

z = (x - μ) / σ = (117.2 - 112) / 2.7 = 1.926

To determine the probability of having a sample mean of 117.2 or less for a random sample of this size = P (x ≤ 117.2) = P (z ≤ 1.926)

We'll use data from the normal probability table for these probabilities

P (x ≤ 117.2) = P (z ≤ 1.926) = 0.973