Statistical Probability and Frequency Distributions

1. The personnel department at Hayes Air West is interested in studying the monthly applicant flow for our entry level service jobs. Find the mean and st. deviation for the following frequency distribution:

Job applicants/month f
78-85 45
86-93 78
94-101 115
102-109 40 110-117 15
118-125 4

Mean:
St. Dev.:

2. Given P(A)=0.47, P(B)=0.35, P(A and B)=0.3
Find: a. P(Ā) b. P(A or B)
c. P(A/B)

3. Find the probability of getting a sum divisable by a three when a pair of dice is thrown.

4. The number of ships to arrive at a harbor on any given day is a random variable represented by x. The probability distribution for x is:

x 20 21 22 23 24
p(x) .15 .25 .20 .15 .25

Find: a. p(exactly 22 ships arrive)
b. p(at most 23 ships arrive)
c. p(µ)
d. p(σ)

5. Given that x is a binomial random variable with n=13 and p=0.8
Find: a. p(x≤6)
b. p(x=12)
c. µ
d. σ

6. A study by the EPA showed that all cars with catalytic converters have a 4.4% chance of having them removed. If 40 cars are a randomly selected, find the probability that exactly 6 cars have had them removed. (binomial probability)

7. Find the following probabilities:
a. p(z<1.49)
b. p(z>-1.87)
c. p(z<-2.46)
d. p(-1.85<Z<0)
e. p(-1.92<Z<2.73)
f. p(-2.57<Z<-1.98)

8. The middle 78% of a normally distributed population lies between what two standard scores?

9. Given a µ=25.1 and &#963;=2.5, find the probability:
a. p(22.4<X<25.9)
b. p(x>27)

10. Lynda's Boutique monthly phone bills are normally distributed with a pop. mean of $73.50 and a pop. st. deviation of $8.98.

a. What % of Lynda's bills > $72?
b. What is the probability that a bill chosen at random < $69?
c. 80% of her bills are below what amount?

11. The length of life of a washing machine is normally distributed with µ=3.5 yrs., and &#963;=0.62 yrs. If the machine is guaranteed for 3 years, what is the probability that the company will be required to replace it?

12. The average length of time required for completing a certain academic test is 150 minutes with a st. deviation of 40 minutes. If we wish to allow sufficient time for only 40% to complete the test, when should the test be terminated? (assume normal distribution)

13. In a large factory, the maintainance dept. has been instructed to replace light bulbs before they burn out. The length of light bulbs is normally distributed with a µ=870 hrs. and &#963;=55hrs. When should the light bulbs be replaced so that no more than 4% of them will burn out while in use?

14. The marketing firm wished to determine whether the number of TV commercials broadcast was linearly correlated to the sales of it's product.
The data for various cities is given to be:
# of comm., x: 11 6 9 15 12 15 8 16 12
Sales units(x10,000),y:
8 5 7 14 12 9 7 11 10

a. What is the value for r?
b. Is there a linear correlation? Why or why not?
c. Write the prediction equation, &#375;=
d. Predict sales when 6 commercials are aired.
e. Predict # of commercials aired if sales were $75,000.

15. A sample of 80 night school ages is obtained. It is known that the average age of a night school student is 32.6 with a st. deviation of 4.9. What is the probability that the sample mean is between 30 and 35?

16. A sample of 60 students is taken in order to estimate the mean travel distance that students travel to school. The mean distance of the sample is 24.6 and st. dev. is 3.96. At the 90% level of confidence, find an interval estimate for the population mean.

17. How large a sample should be taken if the population mean is to be estimated with 95% confidence to within $150? The population st. dev. is $850.

18. In a study on time management, 15 workers were asked to keep track of the time they spent working on a particular kind of part. The average (mean) was 6.1 hours with a st. dev. of 1.2 hours. Use this sample data to construct the 90% confidence interval for the mean time spent on this part.

19. The manager at Air Express feels that the weight of packages recently shipped are greater than in the past. Records show that in the past, the mean weight has been 37.7lbs., with a st. dev. of 8.2lbs. A random sample of last month's shipping records yielded a mean weight of 42.7lbs. for 80 packages. Is this sufficient evidence to support the manager's claim? Use an &#940;=0.05

20. In a large supermarket, the customer's waiting time to check out is approx. norm. dist. With a &#963;=2.2 minutes. A sample of 22 customers produced a mean of 9.6 minutes. Is this evidence sufficient to reject the supermarket's claim that its customers average checkout time is 8 minutes or less? Use &#940;=0.05

21. An educational testing company has been using a standard test of verbal ability with a mean of 430 and st. dev. of 25. In analyzing a new version, it was found that a sample of 150 subjects produced a mean of 400. At the 0.01 level of significance, test the claim that the new version has a mean equal to that of the past version.