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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
(a) What proportion of these components exceed 10150 kg/cm 2 in tensile strength.
(b) If specifications require that all components have tensile strength between 9800 and 10200
kg/cm 2 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
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
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
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.