Answer:
(a) The distribution of <em>X</em> is <em>N</em> (100, 15²).
(b) The probability that a person has an IQ greater than 130 is 0.0228.
(c) The minimum IQ needed to qualify for the Mensa organization is 131.
(d) The middle 20% of IQs fall between 96 and 104.
Step-by-step explanation:
The random variable <em>X</em> is defined as the IQ of an individual.
(a)
The random variable <em>X</em> is normally distributed with mean, <em>μ</em> = 100 and standard deviation, <em>σ</em> = 15.
The probability density function of <em>X</em> is:
![f_{X}(x)=\frac{1}{15\sqrt{2\pi}}\times e^{-(x-100)^{2}/(2\times 15^{2})};\ -\infty](https://tex.z-dn.net/?f=f_%7BX%7D%28x%29%3D%5Cfrac%7B1%7D%7B15%5Csqrt%7B2%5Cpi%7D%7D%5Ctimes%20e%5E%7B-%28x-100%29%5E%7B2%7D%2F%282%5Ctimes%2015%5E%7B2%7D%29%7D%3B%5C%20-%5Cinfty%3CX%3C%5Cinfty)
Thus, the distribution of <em>X</em> is <em>N</em> (100, 15²).
(b)
Compute the probability that a person has an IQ greater than 130 as follows:
![P(X>130)=P(\frac{X-\mu}{\sigma}>\frac{130-100}{15})](https://tex.z-dn.net/?f=P%28X%3E130%29%3DP%28%5Cfrac%7BX-%5Cmu%7D%7B%5Csigma%7D%3E%5Cfrac%7B130-100%7D%7B15%7D%29)
![=P(Z>2)\\=1-P(Z](https://tex.z-dn.net/?f=%3DP%28Z%3E2%29%5C%5C%3D1-P%28Z%3C2%29%5C%5C%3D1-0.97725%5C%5C%3D0.02275%5C%5C%5Capprox%200.0228)
Thus, the probability that a person has an IQ greater than 130 is 0.0228.
(c)
Let <em>x</em> represents the top 2% of all IQs.
Then, P (X > x) = 0.02.
⇒ P (X < x) = 1 - 0.02
⇒ P (Z < z) = 0.98
The value of <em>z</em> is:
<em>z</em> = 2.06.
Compute the value of <em>x</em> as follows:
![z=\frac{x-\mu}{\sigma}\\2.06=\frac{x-100}{15}\\x=100+(2.06\times 15)\\x=130.9\\x\approx131](https://tex.z-dn.net/?f=z%3D%5Cfrac%7Bx-%5Cmu%7D%7B%5Csigma%7D%5C%5C2.06%3D%5Cfrac%7Bx-100%7D%7B15%7D%5C%5Cx%3D100%2B%282.06%5Ctimes%2015%29%5C%5Cx%3D130.9%5C%5Cx%5Capprox131)
Thus, the minimum IQ needed to qualify for the Mensa organization is 131.
(d)
Let <em>x</em>₁ and <em>x</em>₂ be the values between which middle 20% of IQs fall.
This implies that:
![P(x_{1}](https://tex.z-dn.net/?f=P%28x_%7B1%7D%3CX%3Cx_%7B2%7D%29%3D0.20%5C%5CP%28-z%3CZ%3Cz%29%3D0.20%5C%5CP%28Z%3Cz%29-P%28Z%3C-z%29%3D0.20%5C%5CP%28Z%3Cz%29-%5B1-P%28Z%3Cz%29%5D%3D0.20%5C%5C2P%28Z%3Cz%29%3D1.20%5C%5CP%28Z%3Cz%29%3D0.60)
The value of <em>z</em> is:
<em>z</em> = 0.26.
Compute the value of <em>x</em> as follows:
![z=\frac{x_{2}-\mu}{\sigma}\\0.26=\frac{x_{2}-100}{15}\\x_{2}=100+(0.26\times 15)\\x=103.9\\x\approx104](https://tex.z-dn.net/?f=z%3D%5Cfrac%7Bx_%7B2%7D-%5Cmu%7D%7B%5Csigma%7D%5C%5C0.26%3D%5Cfrac%7Bx_%7B2%7D-100%7D%7B15%7D%5C%5Cx_%7B2%7D%3D100%2B%280.26%5Ctimes%2015%29%5C%5Cx%3D103.9%5C%5Cx%5Capprox104)
Thus, the middle 20% of IQs fall between 96 and 104.