EXERCISES FOR PRACTICE SESSION 2 OF STAT CAMP

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1 EXERCISES FOR PRACTICE SESSION 2 OF STAT CAMP Note 1: The exercises below that are referenced by chapter number are taken or modified from the following open-source online textbook that was adapted by Saylor Academy in 2012 under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License without attribution as requested by the work's original creator or licensor: Note 2: You can cut-and-paste into SPSS and Excel column data in any exercises below. 1. (Chapter 3.3, Exercise 2) For two events A and B, P(A) = 0.26, P(B) = 0.37, and P(A B) = a. Find P(A B). b. Find P(B A). c. Determine whether or not A and B are independent. 2. (Chapter 3.3, Exercise 12) A random experiment gave rise to the probabilities in the following table. Use it to compute the probabilities indicated. R S A B a. P(A), P(R), P(A R). b. Based on the answer to (a) and without using Baye s Theorem, determine whether or not the events A and R are independent. c. Based on the answer to (b), determine whether or not P(A R) can be predicted without any computation. If so, make the prediction. In any case, compute P(A R) using Baye s Theorem). 3. (Chapter 4.1, Exercise 2) Classify each random variable as either discrete or continuous. a. The time between customers entering a checkout lane at a retail store. b. The weight of refuse on a truck arriving at a landfill. c. The number of passengers in a passenger vehicle on a highway at rush hour. d. The number of clerical errors on a medical chart. e. The number of accident-free days in one month at a factory. 4. (Chapter 4.1, Exercise 6) Identify the set of possible values for each random variable. a. The number of hearts in a five-card hand drawn from a deck of 52 cards that contains 13 hearts in all. b. The amount of money you might have at the end of playing a game 2 times in which you start with $10 and either double your money or lose your money each time you play, betting $5 each time. c. The number of months in a year that unemployment has decreased. d. The distance a rental car is driven in one 24-hour period at a maximum speed of 80 km per hour.

2 5. (Chapter 4.2, Exercise 4) A discrete random variable X has the following probability distribution: x P(X = x) Compute each of the following quantities. a. P(X = 18). b. P(X > 18). c. P(X 18). d. The mean μ of X. e. The variance σ 2 of X. f. The standard deviation σ of X. 6. (Chapter 4.2, Exercise 12) Seven thousand lottery tickets are sold for $5 each. One ticket will win $2,000, two tickets will win $750 each, and five tickets will win $100 each. Let X denote the amount won from the purchase of a randomly selected ticket. a. Construct the probability distribution of X. b. Compute the expected value of X, namely, E[X]. Interpret E[X] in the context of this problem. c. Compute the standard deviation σ of X. 7. (Chapter 4.2, Exercise 16) An insurance company estimates that the probability that an individual in a particular risk group will survive one year is Such a person wishes to buy a $75,000 one-year term life insurance policy. Let X denote the amount of money the company has at the end of the year when C is how much the insurance company charges such a person for such a policy. a. Construct the probability distribution of X in terms of C. b. Compute the expected value E(X) in terms of C. c. Determine the value C must have in order for the company to break even on all such policies (that is, to average a net gain of zero per policy on such policies). d. Determine the value C must have in order for the company to average a net gain of $150 per policy on all such policies. 8. (Chapter 4.3, Exercise 20) Adverse growing conditions have caused 5% of grapefruits grown in a certain region to be of inferior quality. Grapefruits are sold by the dozen. a. Find the average number of inferior quality grapefruits per box of a dozen. b. A box that contains two or more grapefruits of inferior quality will cause a strong adverse customer reaction. Use Excel to find the probability that a box of one dozen grapefruits will contain two or more grapefruits of inferior quality. 9. (Chapter 4.3, Exercise 22) One-third of all patients who undergo a non-invasive but unpleasant medical test require a sedative. A laboratory performs 20 such tests daily. Let X denote the number of patients on any given day who require a sedative. a. Verify that X satisfies the conditions for a binomial random variable, and find n and p.

3 b. Find the probability that on any given day between five and nine patients will require a sedative (include five and nine). c. Find the average number of patients each day who require a sedative. d. Using the cumulative probability distribution for X in Excel, find the minimum number of doses of the sedative that should be on hand at the start of the day so that there is a 99% chance that the laboratory will not run out. 10. The driving distance, D, for the top 60 women golfers on the LPGA tour is between approximately 239 and 261 yards (Golfweek, December 6, 1997). Answer the following questions assuming that the driving distance for these women is uniformly distributed over this interval. a. Draw the probability density function, indicating its height above the x-axis. b. What is the probability the driving distance for one of these women is at least 255 yards? c. What is the probability the driving distance for one of these women is between 245 and 260 yards? d. How many of these women drive the ball at least 250 yards? (5 points) 11. The random variable T = the number of minutes between two customers arriving at a bank during the busiest time of day is estimated to follow an exponential distribution. The density function depends on the value of a number = the average number of customers arriving per minute which, from data analysis, is estimate to be 1/3 and is given by the formula f(x) = e x, as shown in the graph below. In answering the following questions, you can use the Excel function EXPONDIST to compute the area under this density function to the left of a given value t on the x axis by typing =EXPONDIST(t,, TRUE) Density Function of the Exponential f(x) Series x a. What does the shape of the density function tell you about the likelihood of the different possible values occurring?

4 b. What is the area under this density function over the x axis from 0 to. c. What is the probability that the amount of time between two arriving customers is exactly 4 minutes? d. What is the probability that the amount of time between two arriving customers is less than 2 minutes? e. What is the probability that the amount of time between two arriving customers is more than 3 minutes? f. What is the probability that the amount of time between two arriving customers is between 2 and 3 minutes? 12. (Chapter 5.1, Exercise 6) A continuous random variable X has a normal distribution with mean 169. The probability that X takes a value greater than 180 is Use this information and the symmetry of the density function to find the probability that X takes a value less than 158. Sketch the density curve with relevant regions shaded to illustrate the computation. 13. (Chapter 5.3, Exercise 16) A regulation hockey puck must weigh between 5.5 and 6 ounces. The weights X of pucks made by a particular process are normally distributed with mean 5.75 ounces and standard deviation 0.11 ounce. Find the probability that a puck made by this process will meet the weight standard. 14. (Chapter 5.3, Exercise 18) The length of time that the battery in Hippolyta's cell phone will hold enough charge to operate acceptably is normally distributed with mean 25.6 hours and standard deviation 0.32 hour. Hippolyta forgot to charge her phone yesterday, so that at the moment she first wishes to use it today it has been 26 hours 18 minutes since the phone was last fully charged. Find the probability that the phone will operate properly. 15. (Chapter 5.3, Exercise 24) A machine produces large fasteners whose length must be within 0.5 inch of 22 inches. The lengths are normally distributed with mean 22.0 inches and standard deviation 0.17 inch. a. Find the probability that a randomly selected fastener produced by the machine will have an acceptable length. b. The machine produces 20 fasteners per hour. The length of each one is inspected. Assuming lengths of fasteners are independent, find the probability that all 20 will have acceptable length. Hint: There is a binomial random variable here, whose value of p comes from part (a). 16. (Chapter 5.3, Exercise 28) The amount of gasoline X delivered by a metered pump when it registers 5 gallons is a normally distributed random variable. The standard deviation σ of X measures the precision of the pump; the smaller σ is the smaller the variation from delivery to delivery. A typical standard for pumps is that when they show that 5 gallons of fuel has been delivered the actual amount must be between 4.97 and 5.03 gallons (which corresponds to being off by at most about half a cup). Supposing that the mean of X is 5, use Excel to find the largest that σ can be so that P(4.97 < X < 5.03) is at least 0.98.

5 17. (Chapter 5.4, Exercise 12) Heights of women are normally distributed with mean 63.7 inches and standard deviation 2.47 inches. Find and interpret the height that is the 80th percentile. 18. (Chapter 5.4, Exercise 16) The average finishing time among all high school boys in a particular track event in a certain state is 5 minutes 17 seconds. Times are normally distributed with standard deviation 12 seconds. a. The qualifying time in this event for participation in the state meet is to be set so that only the fastest 5% of all runners qualify. Find the qualifying time to the nearest second. (Hint: Watch out for the time units.) b. In the western region of the state the times of all boys running in this event are normally distributed with standard deviation 12 seconds, but with mean 5 minutes 22 seconds. Find the proportion of boys from this region who qualify to run in this event in the state meet. 19. (Chapter 5.4, Exercise 18) Tests of a new light led to an estimated mean life of 1,321 hours and standard deviation of 106 hours. The manufacturer will advertise the lifetime of the bulb using the largest value for which it is expected that 90% of the bulbs will last at least that long. Assuming bulb life is normally distributed, find that advertised value. 20. (Chapter 5.4, Exercise 22) Researchers wish to investigate the overall health of individuals with abnormally high or low levels of glucose in the blood stream. Suppose glucose levels are normally distributed with mean 96 and standard deviation 8.5 mg/dl, and that normal is defined as the middle 90% of the population. Find the interval of normal glucose levels, that is, the interval centered at 96 that contains 90% of all glucose levels in the population. 21. In cells A6 to D6 in the file CoinFlips.xls, the built-in function RAND () is used to generate a uniform random number between 0 and that is used to compute a value of 0 (which corresponds to flipping a coin and getting a tail) or a 1 (which corresponds to flipping a coin and getting a head) with the probability of a tail being given in cell B3. Use this file to perform the following tasks: (i) copy cells A6 to D6 from Row 6 all the way down to Row 105 (so these 100 rows correspond to flipping four coins 100 times); (ii) In Column E, compute the average of the four values in rows 6 to 105. (Think of these as 100 outcomes of a new random variable Y, which is the average of the four binomial random variables in Columns A, B, C, and D.); (iii) You will notice that the values in Column E are 0, 0.25, 0.5, 0.75, or 1, so enter these as the bin values in rows 3 to 7 in Column F. Now answer the following questions. a. Create a histogram of the values in Column E using the bin values in Column F. What do you think is the distribution of the random variable Y? b. What is your estimate of the mean of this distribution? How does this compare with the mean in cell B3 of the four random variables in Columns A, B, C, and D? c. Does the histogram change if you change the probability of getting a tail in cell B3? How is the mean of this distribution related to the value in cell B3? d. What is your conclusion about the distribution of a random variable that is the average of a group of other random variables? Explain.