The amount of time that a certain type of battery

Suppose that X1, X2, X3 are independent with the common probability mass function P{Xi = 0} = .2, P{Xi = 1} = .3, P{Xi = 3} = .5, i = 1, 2, 3 (a) Plot the probability mass function of X2 = X1 + X2 2 . (b) Determine E[X2] and Var(X2). (c) Plot the probability mass function of X3 = X1 + X2 + X3 3 . (d) Determine E[X3] and Var(X3).

If 10 fair dice are rolled, approximate the probability that the sum of the values obtained (which ranges from 10 to 60) is between 30 and 40 inclusive.

Approximate the probability that the sum of 16 independent uniform (0, 1) random variables exceeds 10.

A roulette wheel has 38 slots, numbered 0, 00, and 1 through 36. If you bet 1 on a specified number, you either win 35 if the roulette ball lands on that number or lose 1 if it does not. If you continually make such bets, approximate the probability that (a) you are winning after 34 bets; (b) you are winning after 1,000 bets; (c) you are winning after 100,000 bets. Assume that each roll of the roulette ball is equally likely to land on any of the 38 numbers.

A highway department has enough salt to handle a total of 80 inches of snowfall. Suppose the daily amount of snow has a mean of 1.5 inches and a standard deviation of .3 inches. (a) Approximate the probability that the salt on hand will suffice for the next 50 days. (b) What assumption did you make in solving part (a)? (c) Do you think this assumption is justified? Explain briefly.

Fifty numbers are rounded off to the nearest integer and then summed. If the individual roundoff errors are uniformly distributed between .5 and .5, what is the approximate probability that the resultant sum differs from the exact sum by more than 3?

A six-sided die, in which each side is equally likely to appear, is repeatedly rolled until the total of all rolls exceeds 400. Approximate the probability that this will require more than 140 rolls.

The amount of time that a certain type of battery functions is a random variable with mean 5 weeks and standard deviation 1.5 weeks. Upon failure, it is immediately replaced by a new battery. Approximate the probability that 13 or more batteries will be needed in a year.

The lifetime of a certain electrical part is a random variable with mean 100 hours and standard deviation 20 hours. If 16 such parts are tested, find the probability that the sample mean is (a) less than 104; (b) between 98 and 104 hours.

A tobacco company claims that the amount of nicotine in its cigarettes is a random variable with mean 2.2 mg and standard deviation .3 mg. However, the sample mean nicotine content of 100 randomly chosen cigarettes was 3.1 mg. What is the approximate probability that the sample mean would have been as high or higher than 3.1 if the companys claims were true?

The lifetime (in hours) of a type of electric bulb has expected value 500 and standard deviation 80. Approximate the probability that the sample mean of n such bulbs is greater than 525 when (a) n = 4; (b) n = 16; (c) n = 36; (d) n = 64.

An instructor knows from past experience that student exam scores have mean 77 and standard deviation 15. At present the instructor is teaching two separate classes one of size 25 and the other of size 64. (a) Approximate the probability that the average test score in the class of size 25 lies between 72 and 82. (b) Repeat part (a) for a class of size 64. (c) What is the approximate probability that the average test score in the class of size 25 is higher than that of the class of size 64? (d) Suppose the average scores in the two classes are 76 and 83.Which class, the one of size 25 or the one of size 64, do you think was more likely to have averaged 83?

If X is binomial with parameters n=150, p=.6, compute the exact value of P{X 80} and compare with its normal approximation both (a) making use of and (b) not making use of the continuity correction.

Each computer chip made in a certain plant will, independently, be defective with probability .25. If a sample of 1,000 chips is tested, what is the approximate probability that fewer than 200 chips will be defective?

A club basketball team will play a 60-game season. Thirty-two of these games are against class A teams and 28 are against class B teams. The outcomes of all the games are independent. The team will win each game against a class A opponent with probability .5, and it will win each game against a class B opponent with probability .7. Let X denote its total number of victories in the season. (a) Is X a binomial random variable? (b) Let XA and XB denote, respectively, the number of victories against class A and class B teams. What are the distributions of XA and XB? (c) What is the relationship between XA, XB, and X? (d) Approximate the probability that the team wins 40 or more games.

Argue, based on the central limit theorem, that a Poisson random variable having mean will approximately have a normal distribution with mean and variance both equal to when is large. If X is Poisson with mean 100, compute the exact probability that X is less than or equal to 116 and compare it with its normal approximation both when a continuity correction is utilized and when it is not. The convergence of the Poisson to the normal is indicated in Figure 6.5.

Use the text disk to compute P{X 10} when X is a binomial random variable with parameters n = 100, p = .1. Now compare this with its (a) Poisson and (b) normal approximation. In using the normal approximation, write the desired probability as P{X < 10.5} so as to utilize the continuity correction.

The temperature at which a thermostat goes off is normally distributed with variance 2. If the thermostat is to be tested five times, find (a) P{S2/2 1.8} (b) P{.85 S2/2 1.15} where S2 is the sample variance of the five data values.

In Problem 18, how large a sample would be necessary to ensure that the probability in part (a) is at least .95?

Consider two independent samples the first of size 10 from a normal population having variance 4 and the second of size 5 from a normal population having variance 2. Compute the probability that the sample variance from the second sample exceeds the one from the first. (Hint: Relate it to the F-distribution.)

Twelve percent of the population is left-handed. Find the probability that there are between 10 and 14 left-handers in a random sample of 100 members of this population. That is, find P{10 X 14}, where X is the number of left-handers in the sample.

Fifty-two percent of the residents of a certain city are in favor of teaching evolution in high school. Find or approximate the probability that at least 50 percent of a random sample of size n is in favor of teaching evolution, when (a) n = 10; (b) n = 100; (c) n = 1,000; (d) n = 10,000.

The following table gives the percentages of individuals of a given city, categorized by gender, that follow certain negative health practices. Suppose a random sample of 300 men is chosen. Approximate the probability that (a) at least 150 of them rarely eat breakfast; (b) fewer than 100 of them smoke. Sleeps 6 Hours Rarely Eats Is 20 Percent or or Less per Night Smoker Breakfast More Overweight Men 22.7 28.4 45.4 29.6 Women 21.4 22.8 42.0 25.6 Source: U.S. National Center for Health Statistics, Health Promotion and Disease Prevention.

(Use the table from Problem 23.) Suppose a random sample of 300 women is chosen. Approximate the probability that (a) at least 60 of them are overweight by 20 percent or more; (b) fewer than 50 of them sleep 6 hours or less nightly.

(Use the table from Problem 23.) Suppose random samples of 300 women and of 300 men are chosen. Approximate the probability that more women than men rarely eat breakfast.

The following table uses data concerning the percentages of teenage male and female full-time workers whose annual salaries fall in different salary groupings. Suppose random samples of 1,000 men and 1,000 women were chosen. Use the table to approximate the probability that (a) at least half of the women earned less than $20,000; (b) more than half of the men earned $20,000 or more; (c) more than half of the women and more than half of the men earned $20,000 or more; (d) 250 or fewer of the women earned at least $25,000; (e) at least 200 of the men earned $50,000 or more; (f ) more women than men earned between $20,000 and $24,999. Earnings Range Percentage of Women Percentage of Men $4,999 or less 2.8 1.8 $5,000 to $9,999 10.4 4.7 $10,000 to $19,999 41.0 23.1 $20,000 to $24,999 16.5 13.4 $25,000 to $49,999 26.3 42.1 $50,000 and over 3.0 14.9 Source: U.S. Department of Commerce, Bureau of the Census.

In 1995 the percentage of the labor force that belonged to a union was 14.9. If five workers had been randomly chosen in that year, what is the probability that none of them would have belonged to a union? Compare your answer to what it would be for the year 1945, when an all-time high of 35.5 percent of the labor force belonged to a union.

The sample mean and sample standard deviation of all San Francisco student scores on the most recent Scholastic Aptitude Test examination in mathematics were 517 and 120. Approximate the probability that a random sample of 144 students would have an average score exceeding (a) 507; (b) 517; (c) 537; (d) 550.

The average salary of newly graduated students with bachelors degrees in chemical engineering is $53,600, with a standard deviation of $3,200. Approximate the probability that the average salary of a sample of 12 recently graduated chemical engineers exceeds $55,000.

A certain component is critical to the operation of an electrical systemandmust be replaced immediately upon failure. If the mean lifetime of this type of component is 100 hours and its standard deviation is 30 hours, how many of the components must be in stock so that the probability that the system is in continual operation for the next 2000 hours is at least .95?