Continuous Probability Distributions

21. For a standard normal distribution, find the area under the curve that lies

(a) to the right of z = 1.84 (b) between z = -1.97 and z = 0.86.

22. For a standard normal distribution, find the value of k such that

(a) P(Z > k) = 0.3015 and (b)P(k < Z < -0.18) = 0.4197.

23. Given a normal distribution with µ = 40 and σ = 6, find the value of x that has

(a) 45% of the area to the left (b) 14% of the area to the right.

24. A electrical firm manufacturers light bulbs that have a life, before burn out, that is normally

distributed with mean equal to 800 hours and a standard deviation of 40 hours. Find the probability

that a bulb burns between 778 and 834 hours.

25. The average grade of an exam is 74, and the standard deviation is 7. If 12% of the class are given

A and the grades are curved to follow a normal distribution, what is the lowest possible A and the

highest possible B?

26. the tensile strength of a certain metal component is normally distributed with a mean 10,000

kilograms per square centimeter. Measurements are recorded to the nearest 50 Kgs per square

centimeter.

(a) What proportion of these components exceed 10150 kg/cm2 in tensile strength.

(b) If specifications require that all components have tensile strength between 9800 and 10200

kg/cm2 what proportion of these would we expect to scrap.

27. A multiple choice quiz has 200 questions each with 4 possible answers of which only 1 is correct.

What is the probability that sheer guess work yields from 25 to 30 correct answers for 80 of the 200

problems about which the student has no knowledge.

28. 1/6 of the male freshmen entering a large state school are out of state students. If the students

are assigned at random to the dormitories, 180 to a building, what is the probability that in a given

dormitory at least 1/5 of the students are from out of state.

Tests of Hypotheses

29. It is claimed that an automobile is driven on the average more than 20,000 km/yr. To test this

claim, a random sample of 100 automobile owners are asked to keep a record of the kilometers they

travel. Would you agree with this claim if the random sample showed an average of 23,500 km and a

standard deviation of 3900 km. use a p-value in your conclusion.

30. Past experience indicates that the time for high school seniors to complete a standardized test in

a normal random variable with a mean of 35 minutes. If a random sample of 20 high school senior

took an average of 33.1 minutes to complete this test with a standard deviation of 4.3 minutes, test

the hypothesis at 0.05 level of significance that µ = 35 minutes against the alternative that µ < 35

minutes.

31. A builder claims that heat pumps are installed in 70% of all homes being constructed today in the

city of Richmond. Would you agree with this claim if a random survey of new homes in this city

shows that 8 out of 15 had heat pumps installed? Use a 0.10 level of significance.

32. A commonly prescribed drug for relieving nervous tension is believed to be only 60% effective.

Experimental results with a new drug administrated to a random sample of 100 adults who were

suffering from nervous tension show that 70% received relief. Is this sufficient evidence to conclude

that the new drug is superior to the one commonly prescribed? Use 0.05 level of significance.

33. A new radar is being considered for a certain defense missile system. The system is checked by

experimenting with actual aircraft in which a kill or a no kill is simulated. If in 300 trials 250 kills

occur, accept or reject, at the 0.04 level of significance. The claim that the probability of a kill with

the new system does not exceed the 0.8 probability of the existing device.

34. Past experience indicates that the time required for high school seniors to complete a

standardized test is a normal random variable with standard deviation of 6 minutes. Test the

hypothesis that σ = 6 against the alternative that σ <6 if a random sample of 20 high school seniors

has a standard deviation s= 4.51. Use a 0.05 level of significance.

35. A study was conducted at physical health department to determine if 8 weeks of training truly

reduces the cholesterol level of the participants. A treatment group consisting of 15 people were

given lecture twice a week on how to reduce their cholesterol level. Another group of 18 peoples of

similar age were randomly selected as a control group. All participant cholesterol level were

recorded at the end of 8 week program and are listed below.

Treatment: 129, 131, 154, 172, 115, 126, 175, 191, 122, 238, 159, 156, 176, 175, 126.

Control: 151, 132, 196, 195, 188, 198, 187, 168, 115, 165, 137, 208, 133, 217, 191, 193, 140,

146.

Can we conclude, at the 5% level of significance, that the average cholesterol level has been

reduced due to the program? Make the appropriate test on means.

36. If a can containing 500 nuts is selected at random from each of three different distributors of

mixed nuts and there, respectively, 345, 313 and 359 peanuts in each of the cans, can we conclude

at the 0.01 level of significance that the mixed nuts of the three distributors contain equal

proportions of peanuts.

37. In a study conducted, the steel rods supplied by two different companies were compared. Ten

sample springs were made out of steel rods supplied by each company and the bounciness was

studied. The data are as follows:

Company A: 9.3, 8.8, 6.8, 8.7, 8.5, 6.7, 8.0, 6.5, 9.2, 7.0.

Company B: 11.0, 9.8, 9.9, 10.2, 10.1, 9.7, 11.0, 11.1. 10.2, 9.6.

Can u conclude that there is virtually no difference in means between the steel rods supplied

by the two companies. Use a p-value to reach the conclusion. Should variances be pooled

here.

38. The following data show the number of defects in 100,000 line of codes in a particular type of

software program in U.S and Japan. Is there enough evidence to claim that there is a significant

difference between the programs of two countries. Test on means. Should variance be pooled.

U.S: 48, 39, 42, 52, 40, 48, 52, 52, 53, 48, 52, 55, 43, 46, 48, 52.

Japan: 50, 48, 42, 40, 43, 48, 50, 46, 38, 38, 36, 40, 40, 48, 48, 45.