HW6

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Trine University *

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6933

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Statistics

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Feb 20, 2024

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docx

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Uploaded by GeneralMetal248

Q1 According to Money magazine, Maryland had the highest mean annual household income of any state in 2018 at $75,847 (Time.com website). Assume that annual household income in Maryland follows a normal distribution with a mean of $75,847 and standard deviation of $33,800. a. What is the probability that a household in Maryland has an annual income of $100,000 or more (to 4 decimals)? 0.2389 @ b. What is the probability that a household in Maryland has an annual income of $40,000 or less (to 4 decimals)? 0.1446 @ c. What is the probability that a household in Maryland has an annual income between $50,000 and $70,000 (to 4 decimals)? 0.2089 @ d. What is the annual income of a household in the 90th percentile of annual household income in Maryland (to the nearest dollar)? $ 119111 @ Q2 Motorola used the normal distribution to determine the probability of defects and the number of defects expected in a production process. Assume a production process produces items with a mean weight of 8 ounces. a. The process standard deviation is 0.10, and the process control is set at plus or minus 2 standard deviations. Units with weights less than 7.8 or greater than 8.2 ounces will be classified as defects. What is the probability of a defect (to 4 decimals)? 0.0456 @ In a production run of 1,000 parts, how many defects would be found (to the nearest whole number)? a6 & b. Through process design improvements, the process standard deviation can be reduced to 0.08. Assume the process control remains the same, with weights less than 7.8 or greater than 8.2 ounces being classified as defects. What is the probability of a defect (to 4 decimals)? 0.0124 @ In a production run of 1,000 parts, how many defects would be found (to the nearest whole number)? 12 & c. What is the advantage of reducing process variation, thereby causing a problem limits to be at a greater number of standard deviations from the mean? " It can substantially reduce the number of defects. V | @ Q3 The XO Group Inc. conducted a 2015 survey of 13,000 brides and grooms married in the United States and found that the average cost of a wedding is $29,858 (XO Group website). Assume that the cost of a wedding is normally distributed with a mean of $29,858 and a standard deviation of $5,600. a. What is the probability that a wedding costs less than $20,000 (to 4 decimals)? 0.0392 @ b. What is the probability that a wedding costs between $20,000 and $30,000 (to 4 decimals)? 0.4728 @ c. For a wedding to be among the 5% most expensive, how much would it have to cost (to the nearest whole number)? $ 39070 &

Q4 A random variable is normally distributed with a mean of 4 = 50 and a standard deviation of o = 5. a. Which of the following graphs accurately represents the probability density function? A. Choose the correct option. A v b. What is the probability that the random variable will assume a value between 45 and 55 (to 4 decimals)? 0.6826 c. What is the probability that the random variable will assume a value between 40 and 60 (to 4 decimals)? 0.9544 Q5 The time needed to complete a final examination in a particular college course is normally distributed with a mean of 77 minutes and a standard deviation of 12 minutes. Answer the following questions. a. What is the probability of completing the exam in one hour or less (to 4 decimals)? 0.0783 @ b. What is the probability that a student will complete the exam in more than 60 minutes but less than 75 minutes (to 4 decimals)? 0.3555 @ c. Assume that the class has 60 students and that the examination period is 90 minutes in length. How many students do you expect will be unable to complete the exam in the allotted time (to nearest whole number)? s & Qb6

Males in the Netherlands are the tallest, on average, in the world with an average height of 183 centimeters (cm) (BBC News website). Assume that the height of men in the Netherlands is normally distributed with a mean of 183 cm and standard deviation of 10.5 cm. a. What is the probability that a Dutch male is shorter than 175 cm (to 4 decimals)? 0.2231 @ b. What is the probability that a Dutch male is taller than 195 cm (to 4 decimals)? 0.1265 @ c. What is the probability that a Dutch male is between 173 and 193 cm (to 4 decimals)? 0.6592 @ d. Out of a random sample of 1000 Dutch men, how many would we expect to be taller than 190 cm (rounded to the nearest whole number)? 253 @ Q7 Given that z is a standard normal random variable, compute the following probabilities (to 4 decimals). a. P(z < -1.0) 0.1587 @ b. P(z > —1.0) 0.8413 @ c. P(z > —1.5) 0.9332 @ d. P(z > —2.5) 0.9938 @ e. P(-3<2<0) 0.4987 @ Q8

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Delta Airlines quotes a flight time of 2 hours, 5 minutes for its flights from Cincinnati to Tampa. Suppose we believe that actual flight times are uniformly distributed between 2 hours and 2 hours, 20 minutes. a. Which of the following graphs accurately represents the probability density function for flight time in minutes? L. fo0 T 0.157 0.1+ 0.051 110 120 130 140 X Minutes 0.051 r 1 " Graph #1 v @ b. What is the probability that the flight will be no more than 5 minutes late (to 2 decimals)? 0.5 @ c. What is the probability that the flight will be more than 10 minutes late (to 2 decimals)? 0.25 @ d. What is the expected flight time, in minutes? 130 @ Q9 ETHF Consider the following exponential probability density function. 1 _= f(m)=§e 2 for >0 a. Which of the following is the formula for P(xz < xg)? _x 1 Plx <zp) =e 2 x, 0 2 Plx<zo)=1—e€ = 3 Plx<zg)=1—e ™ |> Formula #2 v | b. Find P(z < 2) (to 4 decimals). 0.6321 c. Find P(xz > 3) (to 4 decimals). 0.0585 d. Find P(xz < 6) (to 4 decimals). 0.9179 e. Find P(2 < & < 6) (to 4 decimals). 0.2179

Q10 Intensive care units (ICUs) generally treat the sickest patients in a hospital. ICUs are often the most expensive department in a hospital because of the specialized equipment and extensive training required to be an ICU doctor or nurse. Therefore, it is important to use ICUs as efficiently as possible in a hospital. According to a 2017 large-scale study of elderly ICU patients, the average length of stay in the ICU is 3.4 days (Critical Care Medicine journal article). Assume that this length of stay in the ICU has an exponential distribution. Do not round intermediate calculations. a. What is the probability that the length of stay in the ICU is one day or less (to 4 decimals)? 0.2548 @ b. What is the probability that the length of stay in the ICU is between two and three days (to 4 decimals)? 0.1415 @ c. What is the probability that the length of stay in the ICU is more than five days (to 4 decimals)? 0.2298 @ Q11 a. Suppose you bid $12,000. What is the probability that your bid will be accepted (to 2 decimals)? 0.36 @ b. Suppose you bid $14,000. What is the probability that your bid will be accepted (to 2 decimals)? 0.76 @ c. What amount should you bid to maximize the probability that you get the property? $ 15200 & d. Suppose that you know someone is willing to pay you $16,000 for the property. You are considering bidding the amount shown in part (c) but a friend suggests you bid $13,100. Which bid will give you the larger expected profit? " Bid $13100 to maximize the expected profit v | 0 What is the expected profit for this bid (to 2 decimals)? $ 1682.0 @ Q12

The electric-vehicle manufacturing company Tesla estimates that a driver who commutes 50 miles per day in a Model S will require a nightly charge time of around 1 hour and 45 minutes (105 minutes) to recharge the vehicle's battery (Tesla company website). Assume that the actual recharging time required is uniformly distributed between 90 and 120 minutes. a. Give a mathematical expression for the probability density function of battery recharging time for this scenario. 1 A fz) = 30 for 90 <z <120 0 elsewhere i for 90 < z < 120 B. flz)={15 T -"= 0 elsewhere i for 90 < z < 105 c. flz)=< 15 - - 0 elsewhere The correct answer is: A v | @ b. What is the probability that the recharge time will be less than 110 minutes (to 3 decimals)? 0.667 @ c. What is the probability that the recharge time required is at least 100 minutes (to 3 decimals)? 0.667 @ d. What is the probability that the recharge time required is between 95 and 110 minutes (to 3 decimals)? 0.500 @ Q13 Do you dislike waiting in line? Supermarket chain Kroger has used computer simulation and information technology to reduce the average waiting time for customers at 2,300 stores. Using a new system called QueVision, which allows Kroger to better predict when shoppers will be checking out, the company was able to decrease average customer waiting time to just 26 seconds (InformationWeek website). Assume that waiting times at Kroger are exponentially distributed. a. Which of the probability density functions of waiting time is applicable at Kroger? a.f(z)=ie%=%e % forz >0 b. f(z) = %e_% = %e_% forz >0 c. f(z) = %e_% = 2—166_2% forz <0 d.f(m)=%e_%=2—lfie% forz >0 " b v @ b. What is the probability that a customer will have to wait between 15 and 30 seconds (to 4 decimals)? 0.2462 @ c. What is the probability that a customer will have to wait more than 2 minutes (to 4 decimals)? 0.0099 @ Q14

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A person must score in the upper 2% of the population on an IQ test to qualify for membership in Mensa, the international high-IQ society. If IQ scores are normally distributed with a mean of 100 and a standard deviation of 15, what score must a person have to qualify for Mensa (to whole number)? 131 @ Q15 According to the National Association of Colleges and Employers, the 2015 average starting salary for new college graduates in health sciences was $51,541. The average starting salary for new college graduates in business was $53,901 (National Association of Colleges and Employers website). Assume that starting salaries are normally distributed and that the standard deviation for starting salaries for new college graduates in health sciences is $11,000. Assume that the standard deviation for starting salaries for new college graduates in business is $15,000. a. What is the probability that a new college graduate in business will earn a starting salary of at least $65,000 (to 4 decimals)? 0.2296 @ b. What is the probability that a new college graduate in health sciences will earn a starting salary of at least $65,000 (to 4 decimals)? 0.1112 @ c. What is the probability that a new college graduate in health sciences will earn a starting salary less than $40,000 (to 4 decimals)? 0.1469 @ d. How much would a new college graduate in business have to earn in order to have a starting salary higher than 99% of all starting salaries of new college graduates in the health sciences (to the nearest whole number)? $ 77171 @ Q16 Alexa is the popular virtual assistant developed by Amazon. Alexa interacts with users using artificial intelligence and voice recognition. It can be used to perform daily tasks such as making to-do lists, reporting the news and weather, and interacting with other smart devices in the home. In 2018, the Amazon Alexa app was downloaded some 2,800 times per day from the Google Play store (AppBrain website). Assume that the number of downloads per day of the Amazon Alexa app is normally distributed with a mean of 2,800 and standard deviation of 860. a. What is the probability there are 2,000 or fewer downloads of Amazon Alexa in a day (to 4 decimals)? 0.1762 @ b. What is the probability there are between 1,500 and 2,500 downloads of Amazon Alexa in a day (to 4 decimals)? 0.2977 @ c. What is the probability there are more than 3,000 downloads of Amazon Alexa in a day (to 4 decimals)? 0.4090 @ d. Assume that Google has designed its servers so there is probability 0.01 that the number of Amazon Alexa app downloads in a day exceeds the servers' capacity and more servers have to be brought online. How many Amazon Alexa app downloads per day are Google's servers designed to handle (to the nearest whole number)? 4800 @ downloads per day

HW6

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