Question

Assume that adults have IQ scores that are normally distributed with a mean of 96.3 and a standard deviation 23.1 Find the first quartile ​, which is the IQ score separating the bottom​ 25% from the top​ 75%. (Hint: Draw a​ graph.)

Answer 1

Solution :

Let X be a random variable which represents the IQ scores of adults.

Given that, X ~ N(96.3, 23.1²)

μ = 96.3 and  σ = 23.1

Let the first quatile is k.

Since, the 25% of all possible observations lie below the first quartile, therefore

P(X < k) = 0.25

We know that if X ~ N(μ, σ²) then,

Using "qnorm" function of R we get, P(Z < -0.6745) = 0.25

Comparing, P(Z < -0.6745) = 0.25 and (1) we get,

Hence, the first quartile is 80.72.

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