Question

The salaries of Pet Detectives are normally distributed with a mean of $32,000 and a standard deviation of $3000. If 100 Pet Detectives are randomly selected, find the probability that their average salary is greater than $32,500. Select the single best answer choice

Answer 1

`P(\bar X> 32500)`

And for this case we can use the z score formula given by:

`z =(\bar X -\mu)/((\sigma)/(√(n)))`

And replacing the info given we got:

`z =(32500-32000)/((3000)/(√(100)))= 1.67`

And for this case we can find the probability with the normal standard table using the complement rule and we got:

`P(z> 1.67) = 1-P(z<1.67) = 1-0.953 = 0.047`

Explanation:

Let X the random variable that represent the salaries of Pet Detectives of a population, and for this case we know the distribution for X is given by:

`X \sim N(32000,3000)`

Where
`\mu=32000` and
`\sigma=3000`

We select a sample size of n =100 Pet Detectives and we want to find the following probability:

`P(\bar X> 32500)`

And for this case we can use the z score formula given by:

`z =(\bar X -\mu)/((\sigma)/(√(n)))`

And replacing the info given we got:

`z =(32500-32000)/((3000)/(√(100)))= 1.67`

And for this case we can find the probability with the normal standard table using the complement rule and we got:

`P(z> 1.67) = 1-P(z<1.67) = 1-0.953 = 0.047`