Probability Distributions and Random Variables, Study notes of Biology

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Unit 5 Test REVIEW - Random Variables

Multiple Choice Identify the choice that best completes the statement or answers the question.

1. Let X = the score out of 5 points on a randomly selected quiz in a chemistry class. The table gives the probability distribution of X with missing probabilities. Value 0 1 2 3 4 5 Probability?????? To make this a valid probability distribution, which one of the following could be the missing probabilities? A) 0.01, 0.11, 0.03, 0.06, -0.09, 0.88 C) 0.00, 0.01, 0.02, 0.87, 0.13, 0. B) 0.09, 0.00, 0.03, 0.12, 0.56, 0.20 D) 0.03, 0.00, -0.13, 0.25, 0.76, 0. 2. Let X = the score out of 10 points on a randomly selected test a statistics class. The table gives the probability distribution of X. Value 0 1 2 3 4 5 6 7 8 9 10 Probability 0.00 0.01 0.02 0.01 0.01 0.03 0.04 0.23 0.39 0.20 0. A probability of 0.08 was calculated. Which one of these statements matches this probability? A) P ( X > 5) C) P ( X 5) B) P ( X < 5) D) P ( X 5) 3. Which of the following random variables are continuous? I. N = amount of gasoline placed into a randomly selected vehicle’s gas tank II. J = the number of oranges slices eaten by each player on a soccer team at half time during a randomly selected game III. K = the number of books on a randomly selected library shelf A) II and III C) I and II B) I only D) I, II and III 4. Which of the following random variables are discrete? I. L = the number of pages in a randomly selected book II. A = the number of leaves on a randomly selected tree III. K = the height of a randomly selected NBA player A) I only C) I and II B) II and III D) I, II and III 5. Let X = the score out of 10 points on a randomly selected test from a semester in a statistics class. The table gives the probability distribution of X. Value 0 1 2 3 4 5 6 7 8 9 10 Probability 0.06 0.20 0.39 0.23 0.04 0.03 0.01 0.01 0.02 0.01 0. Which of the following graphs gives the correct histogram and description of the probability distribution of X? A) The graph is roughly symmetric and has a

C)

The graph is skewed to the left and has a single peak at 8 points.

single peak at 6 points. B) The graph is skewed to the right and has a single peak at 2 points.

D)

The graph is roughly symmetric and has a single peak at 5 points.

6. Let X = the score out of 3 points on a randomly selected pop quiz in a statistics class. The table gives the probability distribution of X. Value 0 1 2 3 Probability 0.01 0.19 0.33 0. Calculate and interpret the expected value of X. A) The average score will be about 2.26. B) If many, many tests are randomly selected, their average score will be about 2.26. C) If a test is randomly selected, the average score will be about 2.26. D) None of these is correct. 7. Which one of the following is not a binomial setting? A) Shuffle a deck of 52 cards. Turn over a card and then place it back into the deck. Do this 22 times. Record the number of red cards you observe. B) Roll a die 33 times. Record the number of fours you observe. C) Randomly select 10 days from last month. Record the number of days rain occurred in your town. D) 8 red marbles and 2 purple marbles are placed in a hat. Randomly select a marble from the hat and then place it back into the hat. Do this 10 times. Record the number of purple marbles you observe. 8. An online retailer claims the probability of one of their packages arriving on time is 0.99. Assume that delivery times for packages are independent of one another. What is the probability that in 12 randomly selected orders, exactly 10 are on time? A) 0.0062 C) 0. B) 0.9940 D) None of these is correct. 9. An online retailer claims the probability of their orders arriving on time is 0.99. Suppose the retailer randomly selects 100 orders. Let X = the number of orders that arrive on time. Calculate and interpret the mean and standard deviation of X. A) If many orders were shipped, we’d expect about 99 of the orders to arrive on time, on average, and the number of on time orders would typically vary from the mean by about 0.99 orders. B) If many orders were shipped, we’d expect about 99 of the orders to arrive on time, on average and the number of on time orders would typically vary from the mean by about 0.995 orders. C) If many orders were shipped, we’d expect the number of on time orders to be within 0. of the true number of on time orders on average and the number of on time orders would typically vary from the mean by about 0.995 orders. D) If many orders were shipped, we’d expect the number of on time orders to be within 99 of the true number of on time orders on average and the number of on time orders would

typically vary from the mean by about 0.995 orders.

10. A bowler believes the probability that she bowls a strike is 0.8. Suppose she bowls 10 times. Let B = the number of strikes. The probability distribution of B is shown here. Strikes 0 1 2 3 4 5 Probability 0.0000 0.0000 0.0001 0.0008 0.0055 0. Strikes 6 7 8 9 10 Probability 0.0881 0.2013 0.3020 0.2684 0. Find the probability that the bowler gets more than 8 strikes. A) 0.3222 C) 0. B) 0.6242 D) 0. 11. The probability distribution of a random variable D is the density curve of a uniform distribution below. Which of the following is correct? I. The height that makes this a valid density curve is 2.5 units. II. The density curve is entirely above the horizontal axis. III. P (0.3  D 0.5) = 0. A) I and II C) II and III B) I and III D) I, II and III 12. Which of the following is correct about the relative positions of the mean and median for a probability distribution of a continuous random variable that is skewed to the right? A) There is no way to determine the relative positions of the mean and median without the actual graph. B) The mean and median will be equal. C) The median will be larger than the mean. D) The mean will be larger than the median. 13. In each of the normal graphs below, the mean and the points that are 1, 2 and 3 standard deviations from the mean are labeled on the horizontal axis. Which of these graphs has a mean of 12.3 and a standard deviation of 1.9? A) C)

B) D) None of these graphs has a mean of 12. and a standard deviation of 1.9.

14. Using Table A, determine which of the following probabilities is correct. I. The probability that Z is less than 1.43 is 0.9236. II. P (195 < Z 2.06) 0. III. P (0.36  Z 1.59) 0. A) I only C) II and III B) I and II D) I, II and III 15. Use Table A to find the value of z for which P ( Z _____) 0.7054. A) z = 0.54 C) z = 0. B) z = -0.54 D) None of these is correct. 16. Without properly working spark plugs, a vehicle will not run. For a specific vehicle, the spark plugs are supposed to have a gap between 3.9mm and 4.3mm. Any spark plugs with gaps larger or smaller than this will fail inspection and be discarded. At the factory, the machine that sets the gap follows a normal distribution with a mean of 4.1mm and standard deviation of 0.075mm. What is the probability that a randomly selected spark plug passes inspection? A) 0.0077 C) 0. B) 0.9923 D) None of these is correct. 17. At a spark plug factory, the machine that sets the gap follows a normal distribution with a mean of 4.1mm and standard deviation of 0.075mm. Find the 40th percentile of gap distances. A) -0.96mm C) 4.08mm B) 4.12mm D) None of these is correct. 18. At a spark plug factory, the machine that sets the gap follows a normal distribution with a mean of 1.5mm and standard deviation of 0.075mm. Find the 25th percentile of gap distances. A) 1.45mm C) -0. B) 1.55mm D) None of these is correct. Short Answer 19. A survey of random students from two small colleges in the same town was conducted. The number of hours spent working per week on campus was recorded for all randomly selected students. Let A = the number of hours spent working per week for a student from College A and B = the number of hours spent working per week for a student from College B. The table gives the probability distribution of A and B. Hours 18 19 20 College A Probability 0.21 0.29. College B Probability 0.14 0.44 0. a. Calculate and interpret the mean of A. b. The mean of B = 19.28 hours. Which mean was larger? What does this mean in context?

c. Calculate and interpret the standard deviation of A. d. The histograms of A and B appear below. The standard deviation of B = 0.69 hours. Which standard deviation was larger? Use the histograms to explain this difference in context. e. Describe the shape of each histogram.

20. Let K be the random variable described by the probability distribution below. a. What must the height of this probability distribution of K be so that it is a valid density curve? b. Discuss the second condition for density curves. Does this probability distribution fulfill this condition? c. Find P (0.8  T 1.1). d. What is the shape of this density curve?

e. What are the relative locations of the mean and median of this density curve? f. Discuss the approximate locations of the mean and median for a right skewed probability distribution. g. Discuss the approximate locations of the mean and median for a left skewed probability distribution. h. Sketch a normal probability distribution with a mean of 11.6 and a standard deviation of 2.3. Label the mean and the points that are 1, 2 and 3 standard deviations from the mean.

21. The random variable X = weight of a randomly selected bag of chips can be modeled by a normal distribution with

= 22.6 ounces and = 0.11 ounces.

a. Use the 68-95-99.7 rule to approximate each of the following: i. The probability that a randomly selected bag contains more than 22.38 ounces of chips. ii. P (22.49  X 22.93) Z-scores are calculated for all bags of chips. Draw a standard normal distribution with the area of interest shaded for each of the following and then use Table A to find the indicated probabilities: b. The probability that Z is less than 1.25. c. P (–1.86  Z 1.35) d. 75% of bags of chips will have a z-score less than what value? e. Find the value of z for which P( Z > ____ ) = 0.1685.

22. A statistics teacher recently gave a test to 124 students. The scores on the test were approximately normally

distributed with = 76.4and = 3.4.

a. Any student who scores lower than a 70 has the opportunity to retake the test. What percent of students

would be eligible to retake the test?

b. What percent of students scored higher than an 80?

c. About how many students scored lower than an 85?

d. What is the IQR?

23. On the first statistics test of the semester in an introductory statistics class, the test scores were approximately

normally distributed with = 83.2 and = 4.9. Let X = the test score of a randomly selected student.

a. What percentage of students scored between an 80 and 90? Find P (80 < X < 90). b. What percentage of students scored higher than a 90?

c. On the most recent test, the test scores were also approximately normally distributed with = 86.9 and = 2.1. On

which test, the first test or this most recent test, was it more likely for a student to score higher than a 90? Explain. d. Students who scored in the lowest 35% on the most recent test have the opportunity to retake the test. What is the cutoff score for students who can retake the test? .

24. Let C = the number of customers to an automobile repair shop on a randomly selected day. The table gives the probability distribution of C. Customers 0 1 2 3 4 5 6 Probability 0.01 0.06 0.08 0.19 0.28 0.29 0. a. Verify that the probability distribution of Y is valid. b. To be profitable, the repair shop needs to have at least four customers per day. What is the probability that the shop is profitable on a randomly selected day? c. Is Y continuous or discrete? Explain briefly.

d. Make a histogram of the probability distribution and describe its shape.

25. Use Table A to calculate each of the following. a. P ( Z 1.45) b. P (–0.89 < Z 1.05) c. P ( Z –1.45) d. Draw a standard normal distribution, shading in the area corresponding to your answers to parts a and c. Discuss the unique relationship between these answers and the shape of the standard normal distribution. For each of the following, find the indicated value of z and draw a standard normal distribution with the area of interest shaded. e. Find the value of z for which P ( Z > ____) = 0.2946. f. Find the value of z for which P ( Z < ____ ) = 0.0032.

Unit 5 Test REVIEW - Random Variables

Answer Section

MULTIPLE CHOICE

1. ANS: B PTS: 1 DIF: Mod OBJ: Learning Target 5.1. BLM: Applying 2. ANS: C PTS: 1 DIF: Mod OBJ: Learning Target 5.1. BLM: Applying 3. ANS: B PTS: 1 DIF: Mod OBJ: Learning Target 5.1. BLM: Understanding 4. ANS: C PTS: 1 DIF: Mod OBJ: Learning Target 5.1. BLM: Understanding 5. ANS: B PTS: 1 DIF: Mod OBJ: Learning Target 5.2. BLM: Understanding 6. ANS: B PTS: 1 DIF: Mod OBJ: Learning Target 5.2. BLM: Applying 7. ANS: C PTS: 1 DIF: Mod OBJ: Learning Target 5.3. BLM: Applying 8. ANS: C PTS: 1 DIF: Easy OBJ: Learning Target 5.3. BLM: Applying 9. ANS: B PTS: 1 DIF: Easy OBJ: Learning Target 5.4. BLM: Applying 10. ANS: D PTS: 1 DIF: Easy OBJ: Learning Target 5.4. BLM: Applying 11. ANS: D PTS: 1 DIF: Easy OBJ: Learning Target 5.5. BLM: Applying 12. ANS: D PTS: 1 DIF: Mod OBJ: Learning Target 5.5. BLM: Understanding 13. ANS: C PTS: 1 DIF: Easy OBJ: Learning Target 5.5. BLM: Understanding 14. ANS: B PTS: 1 DIF: Easy OBJ: Learning Target 5.6. BLM: Applying 15. ANS: B PTS: 1 DIF: Easy OBJ: Learning Target 5.6. BLM: Applying 16. ANS: B PTS: 1 DIF: Easy OBJ: Learning Target 5.7. BLM: Applying 17. ANS: C PTS: 1 DIF: Easy OBJ: Learning Target 5.7. BLM: Applying 18. ANS: A PTS: 1 DIF: Easy OBJ: Learning Target 5.7. BLM: Applying SHORT ANSWER 19. ANS: a. If many, many students from College A are randomly selected, the average number of hours spent working per week will be about 19.30 hours.

b. The mean of A was slightly larger. This means, that on average, students from College A tend to work more hours per week than those students at College B. c. A randomly selected student from College A will typically differ from the mean by about 0.79 hours. d. The standard deviation of A was larger. The histogram for A has more hours spread out further from the mean of 19.30 hours than the histogram for B which has fewer hours spread out further from the mean of 19.28 hours. e. The histogram of A is single peaked at 20 and skewed to the left. The histogram of B is single peaked at 29 and skewed to the left. PTS: 1 DIF: Diff OBJ: Learning Target 5.2.1 | Learning Target 5.2.2 | Learning Target 5.2. BLM: Applying

20. ANS: a. 1.25 units b. The curve is entirely at or above the horizontal axis, indicating that this is a valid density curve. c. 0. d. Uniform. e. The mean and median will be at the same point, 0. f. The mean will be to the right of the median. g. The median will be to the left of the median. h. PTS: 1 DIF: Easy OBJ: Learning Target 5.5.1 | Learning Target 5.5.2 | Learning Target 5.5. BLM: Applying 21. ANS: a.i. 0. a.ii. 0.

b. 0. c. 0. d. z 0. e. z = 0. PTS: 1 DIF: Mod OBJ: Learning Target 5.6.1 | Learning Target 5.6.2 | Learning Target 5.6. BLM: Applying

22. ANS: a. 0. b. 0. c. 123 students d. 4.

PTS: 1 DIF: Diff OBJ: Learning Target 5.7.1 | Learning Target 5.7. BLM: Applying

23. ANS: a. 0. b. 0. c. On the most recent test, 0.0699 of students got an ‘A’ while on the first test, 0.0826 of students got an ‘A’. It was more likely for a student on the first test to get an ‘A’. d. 81. PTS: 1 DIF: Easy OBJ: Learning Target 5.7.1 | Learning Target 5.7. BLM: Applying 24. ANS: a. Each of the probabilities is between 0 and 1. The sum of the probabilities is 1. b. 0.66. c. Y is discrete because Y takes on a finite number of values between 0 and 6. d. The graph is skewed to the left and has a single peak at 5 customers. PTS: 1 DIF: Easy OBJ: Learning Target 5.1.1 | Learning Target 5.1.2 | Learning Target 5.1.3 | Learning Target 5.2. BLM: Applying 25. ANS: a. 0. b. 0. c. 0.

d. and

. These answers are the same due to the symmetric nature of the standard normal distribution. e. z = 0. f. z = -2. PTS: 1 DIF: Mod OBJ: Learning Target 5.6.2 | Learning Target 5.6. BLM: Applying