Question

iq is normally distributed with a mean of 100 and a standard deviation of 15.
a) suppose one individual is randomly chosen. find the probability that this person has an iq greater than 95.
write your answer in percent form. round to the nearest tenth of a percent.
( p(\text{iq greater than } 95)=) (%)
b) suppose one individual is randomly chosen. find the probability that this person has an iq less than 125.
write your answer in percent form. round to the nearest tenth of a percent.
( p(\text{iq less than } 125)=) (%)
c) in a sample of 600 people, how many people would have an iq less than 110?
people
d) in a sample of 600 people, how many people would have an iq greater than 140?
people
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Explanation:

Part (a)

Step 1: Calculate the z - score

The formula for the z - score is $z=\frac{x-\mu}{\sigma}$, where $x = 95$, $\mu=100$, and $\sigma = 15$.
So, $z=\frac{95 - 100}{15}=\frac{- 5}{15}\approx - 0.33$.

Step 2: Find the probability

We want $P(X>95)$, which is equivalent to $P(Z > - 0.33)$. Since the total area under the standard normal curve is 1, $P(Z>-0.33)=1 - P(Z\leq - 0.33)$.
Looking up the value of $P(Z\leq - 0.33)$ in the standard normal table, we get $P(Z\leq - 0.33)\approx0.3707$.
Then $P(Z > - 0.33)=1 - 0.3707 = 0.6293$.
To convert to a percentage, we multiply by 100: $0.6293\times100 = 62.93\%\approx62.9\%$.

Step 1: Calculate the z - score

Using the z - score formula $z=\frac{x-\mu}{\sigma}$ with $x = 125$, $\mu = 100$, and $\sigma=15$.
$z=\frac{125 - 100}{15}=\frac{25}{15}\approx1.67$.

Step 2: Find the probability

We want $P(X < 125)$, which is equivalent to $P(Z < 1.67)$.
Looking up the value of $P(Z < 1.67)$ in the standard normal table, we get $P(Z < 1.67)\approx0.9525$.
Converting to a percentage: $0.9525\times100=95.25\%\approx95.3\%$.

Step 1: Calculate the z - score

Using the z - score formula $z=\frac{x-\mu}{\sigma}$ with $x = 110$, $\mu = 100$, and $\sigma = 15$.
$z=\frac{110 - 100}{15}=\frac{10}{15}\approx0.67$.

Step 2: Find the probability

We want $P(X < 110)$, which is equivalent to $P(Z < 0.67)$.
Looking up the value of $P(Z < 0.67)$ in the standard normal table, we get $P(Z < 0.67)\approx0.7486$.

Step 3: Calculate the number of people

The number of people with IQ less than 110 in a sample of $n = 600$ is $n\times P(X < 110)=600\times0.7486 = 449.16\approx449$.

Answer:

$62.9$

Part (b)