Question

The Wechsler Adult Intelligence Scale (WAIS) is a common "IQ test" for adults. The distribution of WAIS scores for persons over 16 years of age is approximately Normal with mean 110 and standard deviation 13.

The mean value of x⎯⎯⎯x¯ for an SRS of 55 people is _____

The standard deviation of x⎯⎯⎯x¯ (±±0.01) for an SRS of 55 people is ____

The z-score (±±0.01) corresponding to an average WAIS score of 114 for an SRS of 55 people is _____

What is the probability (±±0.0001) that the average WAIS score of an SRS of 55 people is 114 or higher? _____

Answer 1

The mean value of x⎯⎯⎯x¯ for an SRS of 55 people is 110

The standard deviation of x⎯⎯⎯x¯ (±±0.01) for an SRS of 55 people is = 13/sqrt(55) = 1.7529

The z-score (±±0.01) corresponding to an average WAIS score of 114 for an SRS of 55 people is 2.28

Here, μ = 110, σ = 1.7529 and x = 114. We need to compute P(X >= 114). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (114 - 110)/1.7529 = 2.28

the probability (±±0.0001) that the average WAIS score of an SRS of 55 people is 114 or higher is 0.0113

Therefore,
P(X >= 114) = P(z <= (114 - 110)/1.7529)
= P(z >= 2.28)
= 1 - 0.9887 = 0.0113