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 0.6911. (round to four decimal places as needed.) (c) the probability that a randomly selected utility bill is more than $130 is . (round to four decimal places as needed.)

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. Given $\mu = 100$ and $\sigma=12$.

Step2: Calculate z - score for part (c)

For $x = 130$, $z=\frac{130 - 100}{12}=\frac{30}{12}=2.5$.

Step3: Use the standard normal distribution table

The standard - normal distribution table gives the cumulative probability $P(Z\lt z)$. We want $P(Z > 2.5)$. Since the total area under the standard - normal curve is 1, $P(Z>z)=1 - P(Z < z)$. Looking up $z = 2.5$ in the standard - normal table, $P(Z < 2.5)=0.9938$. So $P(Z>2.5)=1 - 0.9938 = 0.0062$.

Answer:

0.0062