Elementary Statistics: A Step By Step Approach
9th Edition
ISBN: 9780073534985
Author: Allan Bluman
Publisher: McGraw-Hill Science/Engineering/Math
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Chapter 5.2, Problem 6E
Traffic Accidents The county highway department recorded the following
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30 Explain how you can use the empirical rule
to find out whether a data set is mound-
shaped, using only the values of the data
themselves (no histogram available).
5. Let X be a positive random variable with finite variance, and let A = (0, 1).
Prove that
P(X AEX) 2 (1-A)² (EX)²
EX2
6. Let, for p = (0, 1), and xe R. X be a random variable defined as follows:
P(X=-x) = P(X = x)=p. P(X=0)= 1-2p.
Show that there is equality in Chebyshev's inequality for X.
This means that Chebyshev's inequality, in spite of being rather crude, cannot be
improved without additional assumptions.
Chapter 5 Solutions
Elementary Statistics: A Step By Step Approach
Ch. 5.1 - Dropping College Courses Use the following table...Ch. 5.1 - Define and give three examples of a random...Ch. 5.1 - Explain the difference between a discrete and a...Ch. 5.1 - Give three examples of a discrete random variable.Ch. 5.1 - Give three examples of a continuous random...Ch. 5.1 - List three continuous random variables and three...Ch. 5.1 - What is a probability distribution? Give an...Ch. 5.1 - Prob. 7ECh. 5.1 - For Exercises 7 through 12, determine whether the...Ch. 5.1 - Prob. 9E
Ch. 5.1 - For Exercises 7 through 12, determine whether the...Ch. 5.1 - Prob. 11ECh. 5.1 - For Exercises 7 through 12, determine whether the...Ch. 5.1 - Prob. 13ECh. 5.1 - For Exercises 13 through 18, state whether the...Ch. 5.1 - Prob. 15ECh. 5.1 - For Exercises 13 through 18, state whether the...Ch. 5.1 - Prob. 17ECh. 5.1 - For Exercises 13 through 18, state whether the...Ch. 5.1 - Prob. 19ECh. 5.1 - For Exercises 19 through 26, construct a...Ch. 5.1 - Prob. 21ECh. 5.1 - For Exercises 19 through 26, construct a...Ch. 5.1 - For Exercises 19 through 26, construct a...Ch. 5.1 - For Exercises 19 through 26, construct a...Ch. 5.1 - For Exercises 19 through 26, construct a...Ch. 5.1 - For Exercises 19 through 26, construct a...Ch. 5.1 - Triangular Numbers The first six triangular...Ch. 5.1 - Prob. 28ECh. 5.1 - Goals in Hockey The probability that a hockey team...Ch. 5.1 - Mathematics Tutoring Center At a drop-in...Ch. 5.1 - For Exercises 31 through 36, write the...Ch. 5.1 - For Exercises 31 through 36, write the...Ch. 5.1 - For Exercises 31 through 36, write the...Ch. 5.1 - For Exercises 31 through 36, write the...Ch. 5.1 - For Exercises 31 through 36, write the...Ch. 5.1 - For Exercises 31 through 36, write the...Ch. 5.1 - Computer Games The probability that a child plays...Ch. 5.2 - Radiation Exposure On March 28, 1979, the nuclear...Ch. 5.2 - Prob. 1ECh. 5.2 - Suit Sales The number of suits sold per day at a...Ch. 5.2 - Prob. 3ECh. 5.2 - Trivia Quiz The probabilities that a player will...Ch. 5.2 - Prob. 5ECh. 5.2 - Traffic Accidents The county highway department...Ch. 5.2 - Prob. 7ECh. 5.2 - Benfords Law The leading digits in actual data,...Ch. 5.2 - Prob. 9ECh. 5.2 - Pizza Deliveries A pizza shop owner determines the...Ch. 5.2 - Prob. 11ECh. 5.2 - Job Bids A landscape contractor bids on jobs where...Ch. 5.2 - Rolling Dice If a person rolls doubles when she...Ch. 5.2 - Dice Game A person pays 2 to play a certain game...Ch. 5.2 - Lottery Prizes A lottery offers one 1000 prize,...Ch. 5.2 - Winning the Lottery For a daily lottery, a person...Ch. 5.2 - Life Insurance A 35-year-old woman purchases a...Ch. 5.2 - Roulette A roulette wheel has 38 numbers, 1...Ch. 5.2 - Rolling Dice Construct a probability distribution...Ch. 5.2 - Rolling a Die When one die is rolled, the expected...Ch. 5.2 - The formula for finding the variance for a...Ch. 5.2 - Complete the following probability distribution if...Ch. 5.2 - Probability Distribution A bag contains five balls...Ch. 5.3 - Unsanitary Restaurants Health officials routinely...Ch. 5.3 - Which of the following are binomial experiments or...Ch. 5.3 - Which of the following are binomial experiments or...Ch. 5.3 - Compute the probability of X successes, using...Ch. 5.3 - Compute the probability of X successes, using...Ch. 5.3 - Compute the probability of X successes, using the...Ch. 5.3 - Compute the probability of X successes, using the...Ch. 5.3 - Prob. 7ECh. 5.3 - Multiple-Choice Exam A student takes a...Ch. 5.3 - Prob. 9ECh. 5.3 - High School Dropouts Approximately 10.3% of...Ch. 5.3 - Prob. 11ECh. 5.3 - Prob. 12ECh. 5.3 - Prob. 13ECh. 5.3 - Destination Weddings Twenty-six percent of couples...Ch. 5.3 - People Who Have Some College Education Fifty-three...Ch. 5.3 - Guidance Missile System A missile guidance system...Ch. 5.3 - Find the mean, variance, and standard deviation...Ch. 5.3 - Find the mean, variance, and standard deviation...Ch. 5.3 - Prob. 19ECh. 5.3 - Tossing Coins Find the mean, variance, and...Ch. 5.3 - Prob. 21ECh. 5.3 - Federal Government Employee E-mail Use It has been...Ch. 5.3 - Watching Fireworks A survey found that 21% of...Ch. 5.3 - Alternate Sources of Fuel Eighty-five percent of...Ch. 5.3 - Survey on Bathing Pets A survey found that 25% of...Ch. 5.3 - Survey on Answering Machine Ownership In a survey,...Ch. 5.3 - Poverty and the Federal Government One out of...Ch. 5.3 - Internet Purchases Thirty-two percent of adult...Ch. 5.3 - Prob. 29ECh. 5.3 - Job Elimination In a recent year, 13% of...Ch. 5.3 - Survey of High School Seniors Of graduating high...Ch. 5.3 - Is this a binomial distribution? Explain.Ch. 5.3 - Children in a Family The graph shown here...Ch. 5.3 - Construct a binomial distribution graph for the...Ch. 5.3 - Show that the mean for a binomial random variable...Ch. 5.4 - Prob. 1ACCh. 5.4 - Use the multinomial formula and find the...Ch. 5.4 - Use the multinomial formula and find the...Ch. 5.4 - MMs Color Distribution According to the...Ch. 5.4 - Truck Inspection Violations The probabilities are...Ch. 5.4 - Prob. 5ECh. 5.4 - Mendels Theory According to Mendels theory, if...Ch. 5.4 - Prob. 7ECh. 5.4 - Find each probability P(X; ) using Table C in...Ch. 5.4 - Study of Robberies A recent study of robberies for...Ch. 5.4 - Misprints on Manuscript Pages In a 400-page...Ch. 5.4 - Colors of Flowers A nursery provides red impatiens...Ch. 5.4 - Mail Ordering A mail-order company receives an...Ch. 5.4 - Company Mailing Of a companys mailings 1.5% are...Ch. 5.4 - Emission Inspection Failures If 3% of all cars...Ch. 5.4 - Phone Inquiries The average number of phone...Ch. 5.4 - Defective Calculators In a batch of 2000...Ch. 5.4 - School Newspaper Staff A school newspaper staff is...Ch. 5.4 - Job Applicants Twelve people apply for a teaching...Ch. 5.4 - Prob. 19ECh. 5.4 - Defective Computer Keyboards A shipment of 24...Ch. 5.4 - Defective Electronics A shipment of 24 smartphones...Ch. 5.4 - Job Applications Ten people apply for a job at...Ch. 5.4 - Prob. 23ECh. 5.4 - Prob. 24ECh. 5.4 - Shooting an Arrow Mark shoots arrows at a target...Ch. 5.4 - Amusement Park Game At an amusement park...Ch. 5.4 - Prob. 27ECCh. 5.4 - Prob. 28ECCh. 5.4 - Drawing Cards Cards are drawn at random from a...Ch. 5.4 - Prob. 30ECCh. 5.4 - Prob. 31ECCh. 5.4 - Lessons Outside of School About 2 out of every 3...Ch. 5.4 - Prob. 33ECCh. 5.4 - Work versus Conscience One worker in four in...Ch. 5 - For Exercises 1 through 3, determine whether the...Ch. 5 - For Exercises 1 through 3, determine whether the...Ch. 5 - Prob. 5.1.3RECh. 5 - Prob. 5.1.4RECh. 5 - Credit Cards A large retail company encourages its...Ch. 5 - Prob. 5.1.6RECh. 5 - Prob. 5.1.7RECh. 5 - Prob. 5.2.8RECh. 5 - Arrivals at an Airport At a small rural airport,...Ch. 5 - Cans of Paint Purchased During a recent paint sale...Ch. 5 - Prob. 5.2.11RECh. 5 - Outdoor Regatta A producer plans an outdoor...Ch. 5 - Card Game A game is set up as follows: All the...Ch. 5 - Card Game Using Exercise 13, how much should be...Ch. 5 - Let x be a binomial random variable with n = 12...Ch. 5 - Internet Access via Cell Phone In a retirement...Ch. 5 - Prob. 5.3.17RECh. 5 - Prob. 5.3.18RECh. 5 - U.S. Police Chiefs and the Death Penalty The...Ch. 5 - Prob. 5.3.20RECh. 5 - Pizza for Breakfast Three out of four American...Ch. 5 - Prob. 5.3.22RECh. 5 - Prob. 5.4.23RECh. 5 - Prob. 5.4.24RECh. 5 - Prob. 5.4.25RECh. 5 - Prob. 5.4.26RECh. 5 - Prob. 5.4.27RECh. 5 - Boating Accidents The number of boating accidents...Ch. 5 - Prob. 5.4.29RECh. 5 - Prob. 5.4.30RECh. 5 - Items Donated to a Food Bank At a food bank a case...Ch. 5 - Prob. 5.4.32RECh. 5 - Prob. 5.4.33RECh. 5 - Determine whether each statement is true or false....Ch. 5 - Determine whether each statement is true or false....Ch. 5 - Determine whether each statement is true or false....Ch. 5 - Determine whether each statement is true or false....Ch. 5 - Complete these statements with the best answer. 5....Ch. 5 - Complete these statements with the best answer. 6....Ch. 5 - Complete these statements with the best answer. 7....Ch. 5 - Select the best answer. 8. What is the sum of the...Ch. 5 - Select the best answer. 9. How many outcomes are...Ch. 5 - Select the best answer. 10. The number of trials...Ch. 5 - Prob. 11CQCh. 5 - Prob. 12CQCh. 5 - For exercises 11 through 14, determine if the...Ch. 5 - For exercises 11 through 14, determine if the...Ch. 5 - Prob. 15CQCh. 5 - Prob. 16CQCh. 5 - Prob. 17CQCh. 5 - Calls for a Crisis Hot Line The number of calls...Ch. 5 - Selecting a Card There are 6 playing cards placed...Ch. 5 - Prob. 20CQCh. 5 - Carpooling If 40% of all commuters ride to work in...Ch. 5 - Employed Women If 60% of all women are employed...Ch. 5 - Prob. 23CQCh. 5 - Meeting Attendance A history class has 75 members....Ch. 5 - Prob. 25CQCh. 5 - Quality Control Check Before a television set...Ch. 5 - Bowling Team Uniforms Among the teams in a bowling...Ch. 5 - Elm Trees If 8% of the population of trees are elm...Ch. 5 - Sports Score Hot Line Calls Sports Scores Hot Line...Ch. 5 - Color of Raincoats There are 48 raincoats for sale...Ch. 5 - Youth Group Officers A youth group has 8 boys and...Ch. 5 - Blood Types About 4% of the citizens of the United...Ch. 5 - Alcohol Abstainers About 35% of Americans abstain...Ch. 5 - Prob. 1CTCCh. 5 - Prob. 2CTCCh. 5 - Prob. 3CTCCh. 5 - Prob. 4CTCCh. 5 - Prob. 5CTC
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- 4. Prove that, for any random variable X, the minimum of EIX-al is attained for a = med (X).arrow_forward8. Recall, from Sect. 2.16.4, the likelihood ratio statistic, Ln, which was defined as a product of independent, identically distributed random variables with mean 1 (under the so-called null hypothesis), and the, sometimes more convenient, log-likelihood, log L, which was a sum of independent, identically distributed random variables, which, however, do not have mean log 1 = 0. (a) Verify that the last claim is correct, by proving the more general statement, namely that, if Y is a non-negative random variable with finite mean, then E(log Y) log(EY). (b) Prove that, in fact, there is strict inequality: E(log Y) < log(EY), unless Y is degenerate. (c) Review the proof of Jensen's inequality, Theorem 5.1. Generalize with a glimpse on (b).arrow_forward3. Prove that, for any random variable X, the minimum of E(X - a)² is attained for a = EX. Provedarrow_forward
- 7. Cantelli's inequality. Let X be a random variable with finite variance, o². (a) Prove that, for x ≥ 0, P(X EX2x)≤ 02 x² +0² 202 P(|X - EX2x)<≤ (b) Find X assuming two values where there is equality. (c) When is Cantelli's inequality better than Chebyshev's inequality? (d) Use Cantelli's inequality to show that med (X) - EX ≤ o√√3; recall, from Proposition 6.1, that an application of Chebyshev's inequality yields the bound o√√2. (e) Generalize Cantelli's inequality to moments of order r 1.arrow_forwardThe college hiking club is having a fundraiser to buy new equipment for fall and winter outings. The club is selling Chinese fortune cookies at a price of $2 per cookie. Each cookie contains a piece of paper with a different number written on it. A random drawing will determine which number is the winner of a dinner for two at a local Chinese restaurant. The dinner is valued at $32. Since fortune cookies are donated to the club, we can ignore the cost of the cookies. The club sold 718 cookies before the drawing. Lisa bought 13 cookies. Lisa's expected earnings can be found by multiplying the value of the dinner by the probability that she will win. What are Lisa's expected earnings? Round your answer to the nearest cent.arrow_forwardThe Honolulu Advertiser stated that in Honolulu there was an average of 659 burglaries per 400,000 households in a given year. In the Kohola Drive neighborhood there are 321 homes. Let r be the number of homes that will be burglarized in a year. Use the formula for Poisson distribution. What is the value of p, the probability of success, to four decimal places?arrow_forward
- The college hiking club is having a fundraiser to buy new equipment for fall and winter outings. The club is selling Chinese fortune cookies at a price of $2 per cookie. Each cookie contains a piece of paper with a different number written on it. A random drawing will determine which number is the winner of a dinner for two at a local Chinese restaurant. The dinner is valued at $32. Since fortune cookies are donated to the club, we can ignore the cost of the cookies. The club sold 718 cookies before the drawing. Lisa bought 13 cookies. Lisa's expected earnings can be found by multiplying the value of the dinner by the probability that she will win. What are Lisa's expected earnings? Round your answer to the nearest cent.arrow_forwardWhat was the age distribution of nurses in Great Britain at the time of Florence Nightingale? Thanks to Florence Nightingale and the British census of 1851, we have the following information (based on data from the classic text Notes on Nursing, by Florence Nightingale). Note: In 1851 there were 25,466 nurses in Great Britain. Furthermore, Nightingale made a strict distinction between nurses and domestic servants. Use a histogram and graph the probability distribution. Using the graph of the probability distribution determine the probability that a British nurse selected at random in 1851 would be 40 years of age or older. Round your answer to nearest thousandth. Age range (yr) 20–29 30–39 40–49 50–59 60–69 70–79 80+ Midpoint (x) 24.5 34.5 44.5 54.5 64.5 74.5 84.5 Percent of nurses 5.7% 9.7% 19.5% 29.2% 25.0% 9.1% 1.8%arrow_forwardWhat was the age distribution of nurses in Great Britain at the time of Florence Nightingale? Thanks to Florence Nightingale and the British census of 1851, we have the following information (based on data from the classic text Notes on Nursing, by Florence Nightingale). Note: In 1851 there were 25,466 nurses in Great Britain. Furthermore, Nightingale made a strict distinction between nurses and domestic servants. Use a histogram and graph the probability distribution. Using the graph of the probability distribution determine the probability that a British nurse selected at random in 1851 would be 40 years of age or older. Round your answer to nearest thousandth. Age range (yr) 20–29 30–39 40–49 50–59 60–69 70–79 80+ Midpoint (x) 24.5 34.5 44.5 54.5 64.5 74.5 84.5 Percent of nurses 5.7% 9.7% 19.5% 29.2% 25.0% 9.1% 1.8%arrow_forward
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