Question

note: please make sure to properly format your answers. all dollar - figures in the answers need to include the dollar sign and any amount over 1,000 should include the comma ($2,354.67). all percentage values in the answers need to include a percentage sign (%). for all items without specific rounding instructions, round your answers to two decimal places, show both decimal places (5.00).
the distribution of ages of licensed drivers shopping at the kalish county mall is normally distributed with mean 41 and standard deviation 7. a person will be selected at random. round your answers to the nearest hundredth.
a. what is the probability that the person is 48 or older?
b. what is the probability that the person is 48 or younger?
c. what is the probability that the person is between 41 and 48?
d. what is the probability that the person selected will be less than 27 years old?

Explanation:

Step1: Calculate z - score formula

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $x$ is the value from the data set, $\mu$ is the mean and $\sigma$ is the standard deviation. Here, $\mu = 41$ and $\sigma=7$.

Step2: Calculate z - score for $x = 48$

$z=\frac{48 - 41}{7}=\frac{7}{7}=1$.

Step3: Find probability for part a

We want $P(X\geq48)$. Using the standard normal distribution table, $P(Z\geq1)=1 - P(Z < 1)$. From the standard - normal table, $P(Z < 1)=0.8413$, so $P(Z\geq1)=1 - 0.8413 = 0.1587\approx0.16$.

Step4: Find probability for part b

We want $P(X\leq48)$. Since $z = 1$ for $x = 48$, from the standard - normal table, $P(Z\leq1)=0.8413\approx0.84$.

Step5: Find probability for part c

We want $P(41

Step6: Calculate z - score for $x = 27$

$z=\frac{27 - 41}{7}=\frac{-14}{7}=-2$.

Step7: Find probability for part d

We want $P(X < 27)$. Using the standard - normal table, $P(Z < - 2)=0.0228\approx0.02$.

Answer:

a. $0.16$
b. $0.84$
c. $0.34$
d. $0.02$