Question

the monthly utility bills in a city are normally distributed, with a mean of $100 and a standard deviation of $12. find the probability that a randomly selected utility bill is (a) less than $65, (b) between $85 and $110, and (c) more than $130. (a) the probability that a randomly selected utility bill is less than $65 is 0.0018. (round to four decimal places as needed.) (b) the probability that a randomly selected utility bill is between $85 and $110 is (round to four decimal places as needed.)

Explanation:

Step1: Calculate z - scores

The formula for the z - score is $z=\frac{x-\mu}{\sigma}$, where $\mu = 100$ (mean), $\sigma = 12$ (standard deviation), and $x$ is the value from the data set.

Step2: For part (b), find z - scores for $x = 85$ and $x = 110$

For $x = 85$, $z_1=\frac{85 - 100}{12}=\frac{- 15}{12}=-1.25$.
For $x = 110$, $z_2=\frac{110 - 100}{12}=\frac{10}{12}\approx0.83$.

Step3: Use the standard normal distribution table

We know that $P(-1.25From the standard - normal table, $P(Z < 0.83)=0.7967$ and $P(Z < - 1.25)=0.1056$.

Step4: Calculate the probability

$P(-1.25

Answer:

0.6911