Finding Probabilities using the Normal Distribution

1. A national study found that college students with jobs worked an average of 22 hours per week. The standard deviation is 9 hours. A college student with a job is selected at random. Find the probability that the student works for less than 4 hours per week. Assume that the lengths of time college student's work are normally distributed and are represented by the variable X.

Hint: Convert the normal distribution X to Standard normal using Z formula Z = x-n /Q and then look the Z-values from the table and then find the probability. show all your work

2. The average speed of vehicles traveling on a stretch of highway is 67 miles per hour with a standard deviation of 3.5 miles per hour. A vehicle is selected at random. What is the probability that it is violating the speed limit of 70 miles per hour? show your work . Assume the speeds are normally distributed and are represented by the variable X.

Hint: Convert the normal distribution X to Standard normal using Z formula Z = x-n/Q and then look the Z-values from the table and then find the probability.

3. A survey indicates that for each trip to a supermarket, a shopper spends an average of 43 minutes with a standard deviation of 12 minutes in the store. The lengths of time spent in the store are normally distributed and are represented by the variable X. A shopper enters the store.

(a) Find the probability that the shopper will be in the store for each interval of time listed below. show your work

(b) When 200 shoppers enter the store, how many shoppers would you expect to be in the store for each interval of time listed below? show your work

1. Between 33 and 66 minutes.

2. More than 39 minutes

Hint: Convert the normal distribution X to Standard normal using Z formula Z = x-n/Q and then look the Z-values from the table and then find the probability.

4. What is the probability that the shopper in Example 3 will be in the supermarket between 31 and 58 minutes? When 200 shoppers enter the store, how many shoppers would you expect to be in the store between 31 and 58 minutes? show your work .

Hint: Convert the normal distribution X to Standard normal using Z formula Z = x-n/Q and then look the Z-values from the table and then find the probability.

5. Utility Bills The monthly utility bills in a city are normally distributed and represented by the variable X, with a mean of $100 and a standard deviation of $12. show your work. Find the probability that a randomly selected utility bill is

(a) less than $70,

(b) between $90 and $120, and

(c) more than $140.

Hint: Convert the normal distribution X to Standard normal using Z formula Z = x-n/Q and then look the Z-values from the table and then find the probability.

6. Using the Normal Distribution to find the Z-value:

Find the Z-value for the following cumulative areas: show your work

a) A=36.32%

b) A= 10.75%

c) A= 90%

d) A= 95%

e) A= 5%

f) A= 50%

7. From Z-score to random variable X:

Use the below formula X = n + ZQ , then Find X, here X is normally distributed, with a mean 9 and standard deviation 2. show your work

A) Z= -0.5

B) Z= 0

C) Z= 1.96

8. Scores for the California Peace Officer Standards and Training test are normally distributed, with a mean of 50 and a standard deviation of 10. An agency will only hire applicants with scores in the top 5%. What is the lowest score an applicant can earn and still be eligible to be hired by the agency? show your work

9. The lengths of time employees have worked at a corporation are normally distributed, with a mean of 11.2 years and a standard deviation of 2.1 years. In a company cutback, the lowest 10% in seniority are laid off. What is the maximum length of time an employee could have worked and still be laid off? show your work