Question

on a planet far far away from earth, iq of the ruling species is normally distributed with a mean of 103 and a standard deviation of 18. suppose one individual is randomly chosen. let x = iq of an individual. a. what is the distribution of x? x ~ n(103, 18) b. find the probability that a randomly selected persons iq is over 99. round your answer to 4 decimal places. c. a school offers special services for all children in the bottom 6% for iq scores. what is the highest iq score a child can have and still receive special services? round your answer to 2 decimal places. d. find the inter quartile range (iqr) for iq scores. round your answers to 2 decimal places. q1: q3: iqr: enter an integer or decimal number, accurate to at least 4 decimal places more...

Explanation:

Step1: Standardize the value for part b

We use the formula $Z=\frac{X - \mu}{\sigma}$, where $\mu = 103$, $\sigma=18$ and $X = 99$. So $Z=\frac{99 - 103}{18}=\frac{-4}{18}\approx - 0.2222$. We want $P(X>99)=P(Z>-0.2222)$. Since $P(Z > - 0.2222)=1 - P(Z\leq - 0.2222)$. Looking up in the standard - normal table, $P(Z\leq - 0.2222)\approx0.4121$, so $P(Z>-0.2222)=1 - 0.4121 = 0.5879$.

Step2: Find the z - score for part c

We want to find the $z$ - score $z_0$ such that $P(Z\leq z_0)=0.06$. Looking up in the standard - normal table (inverse lookup), $z_0\approx - 1.56$. Then we use the formula $X=\mu+z\sigma$. Substituting $\mu = 103$, $\sigma = 18$ and $z=-1.56$, we get $X=103+(-1.56)\times18=103 - 28.08=74.92$.

Step3: Find Q1 and Q3 for part d

For Q1, we want to find the $z$ - score $z_1$ such that $P(Z\leq z_1)=0.25$. Looking up in the standard - normal table, $z_1\approx - 0.6745$. Then $Q1=\mu+z_1\sigma=103+(-0.6745)\times18=103 - 12.141 = 90.86$.
For Q3, we want to find the $z$ - score $z_3$ such that $P(Z\leq z_3)=0.75$. Looking up in the standard - normal table, $z_3\approx0.6745$. Then $Q3=\mu+z_3\sigma=103 + 0.6745\times18=103+12.141 = 115.14$.
The inter - quartile range $IQR = Q3 - Q1=115.14 - 90.86=24.28$.

Answer:

c. 74.92
d. Q1: 90.86
Q3: 115.14
IQR: 24.28