AP Statistics Review – Probability
AP Statistics Review III– Probability & Random Variables (20% - 30%)
A. PROBABILITY
(THE BIG IDEA: Chance is unpredictable in the _________ run, but follows regular patterns in the _________ run.
|Random |Sample Space |
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|Probability |Event |
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|Complement |Union |Intersection |
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|Disjoint Events |Independent Events |
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( Addition Rule ( Conditional Probability ( Multiplication Rule
B. Random Variables
|Discrete |Continuous |
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|Mean of a Random Variable |Variance of a Random Variable |
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|Binomial R.V. |Geometric R.V. |
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C. Combining Random Variables
Rules for Means: Rules for Variance:
1. 1.
2. 2. If x and y are independent:
3. If x and y are not independent with correlation ρ:
D. The Normal Distribution
D. Normal Distribution
The area under the standard normal curve is:
The standard deviation of the standard normal curve is:
The mean of the standard normal curve is:
What is the rule that describes 1, 2, and 3 standard deviations from the mean?
Name:
IMPORTANT FOR NORMAL DISTRIBUTIONS:
z-score =
The z-score represents
E. Sampling Distributions
Sampling Distribution for the sample mean, [pic] Sampling Distribution for the sample proportion, [pic]
for a sample of size n for a sample of size n
µ p
Multiple Choice Practice
1. A manufacturer makes lightbulbs and claims that their reliability is 98 percent. Reliability is defined to be the proportion of non-defective items that are produced over the long term. If the company’s claim is correct, what is the expected number of non-defective lightbulbs in a random sample of 1,000 bulbs?
(A) 20 (B) 200 (C) 960 (D) 980 (E) 1,000
2. The heights of adult women are approximately normally distributed about a mean of 65 inches with a standard deviation of 2 inches. If Rachael is at the 99th percentile in height for adult women, then her height, in inches, is closest to
(A) 60 (B) 62 (C) 68 (D) 70 (E) 74
3. Gina’s doctor told her that the standardized score (z-score) for her systolic blood pressure, as compared to the blood pressure of other women her age, is 1.50. Which of the following is the best interpretation of this standardized score?
(A) Gina’s systolic blood pressure is 150.
(B) Gina’s systolic blood pressure is 1.50 standard deviations above the average systolic blood pressure of women her age.
(C) Gina’s systolic blood pressure is 1.50 above the average systolic blood pressure of women her age.
(D) Gina’s systolic blood pressure is 1.50 times the average systolic blood pressure for women her age.
(E) Only 1.5% of women Gina’s age have a higher systolic blood pressure than she does.
4. The distribution of the weights of loaves of bread from a certain bakery follows approximately a normal distribution. Based on a very large sample, it was found that 10 percent of the loaves weighed less than 15.34 ounces, and 20 percent of the loaves weighed more than 16.31 ounces. What are the mean and standard deviation of the distribution of the weights of the loaves of bread?
(A) µ = 15.82, σ = 0.48
(B) µ = 15.82, σ = 0.69
(C) µ = 15.87, σ = 0.50
(D) µ = 15.93, σ = 0.46
(E) µ = 16.00, σ = 0.50
5. The weight of adult male grizzly bears living in the wild in the continental United States is approximately normally distributed with a mean of 500 pounds and a standard deviation of 50 pounds. The weight of adult female grizzly bears is approximately normally distributed with a mean of 300 pounds and a standard deviation of 40 pounds. Approximately, what would be the weight of a female grizzly bear with the same standardized score (z-score) as a male grizzly bear with a weight of 530 pounds?
(A) 276 pounds (B) 324 pounds (C) 330 pounds (D) 340 pounds (E) 530 pounds
6. A company sells concrete in batches of 5 cubic yards. The probability distribution of X, the number of cubic yards sold in a single order for concrete from this company, is shown in the table below.
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The expected value of the probability distribution of X is 19.25 and the standard deviation is 5.76. There is a fixed cost to deliver the concrete. The profit Y, in dollars, for a particular order can be described by
Y = 75X – 100. What is the standard deviation Y ?
(A) $332.00 (B) $432.00 (C) $532.00 (D) $1,343.75 (E) $1,400.00
7. A complex electronic device contains three components, A, B, and C. The probabilities of failure for each component in any one year are 0.01, 0.03, and 0.04, respectively. If any one component fails, the device will fail. If the components fail independently of one another, what is the probability that the device will not fail in one year?
(A) Less than 0.01 (B) 0.078 (C) 0.080 (D) 0.922 (E) Greater than 0.99
8. The commuting time for a student to travel from home to a college campus is normally distributed with a mean of 30 minutes and a standard deviation of 5 minutes. If the student leaves home at 8:25 A.M., what is the probability that the student will arrive at the college campus later than 9 A.M. ?
(A) 0.16 (B) 0.32 (C) 0.50 (D) 0.84 (E) 1.00
9. There were 5,317 previously owned homes sold in a western city in the year 2000. The distribution of the sales prices of these homes was strongly right-skewed, with a mean of $206,274 and a standard deviation of $37,881. If all possible simple random samples of size 100 are drawn from this population and the mean is computed for each of these samples, which of the following describes the sampling distribution of the sample mean?
(A) Approximately normal with mean $206,274 and standard deviation $3,788
(B) Approximately normal with mean $206,274 and standard deviation $37,881
(C) Approximately normal with mean $206,274 and standard deviation $520
(D) Strongly right-skewed with mean $206,274 and standard deviation $3,788
(E) Strongly right-skewed with mean $206,274 and standard deviation $37,881
Free Response Practice
10. (2017 #3) A grocery store purchases melons from two distributors, J and K. Distributor J provides melons from organic farms. The distribution of the diameters of the melons from Distributor J is approximately normal with mean 133 millimeters (mm) and standard deviation 5 mm.
(a) For a melon selected at random from Distributor J, what is the probability that the melon will have a diameter greater than 137 mm?
Distributor K provides melons from nonorganic farms. The probability is 0.8413 that a melon selected at random from Distributor K will have a diameter greater than 137 mm. For all the melons at the grocery store, 70 percent of the melons are provided by Distributor J and 30 percent are provided by Distributor K.
(b) For a melon selected at random from the grocery store, what is the probability that the melon will have a diameter greater than 137 mm?
(c) Given that a melon selected at random from the grocery store has a diameter greater than 137 mm, what is the probability that the melon will be from Distributor J.
11. (2016 #4) A company manufactures model rockets that require igniters to launch. Once an igniter is used to launch a rocket, the igniter cannot be reused. Sometimes an igniter fails to operate correctly, and the rocket does not launch. The company estimates that the overal failure rate, defined as the percent of all igniters that fail to operate correctly, is 15 percent.
A company engineer develops a new igniter, called the super igniter, with the intent of lowering the failure rate. To test the performance of the super igniters, the engineer uses the following process.
Step 1: One super igniter is selected at random and used in a rocket.
Step 2: If the rocket launches, another super igniter is selected at random and used in a rocket.
Step 2 is repeated until the process stops. The process stops when a super igniter fails to operate correctly or 32 super igniters have successfully launched rockets, whichever comes first. Assume that super igniter failures are independent.
(a) If the failure rate of the super igniters is 15 percent, what is the probability that the first 30 super igniters selected using the testing process successfully launch rockets?
(b) Given that the first 30 super igniters launch rockets, what is the probability that the first failure occurs on the thirty-first or the thirty-second super ignititer tested if the failure rate of the super igniters is 15 percent?
(c) Given that the first 30 super igniters successfully launch rockets, is it reasonable to believe that the failure rate of the super ignitiers is less than 15 percent? Explain.
12. Of the 10,000 freshmen at the University of Texas, 7000 must take English, 6000 must take History, and 5000 must take both. Suppose that a student is randomly selected.
(a) What is the probability that the selected student must take English?
(b) What is the probability that the selected student must take both English and History?
(c) Suppose you learn that the selected student must take English. What is the probability that this student must also take History?
(d) Are the outcomes must take English and must take History independent? Explain.
(e) Answer the question posed in part (d) if only 4200 of the students must take both English and History.
13. A charity fundraiser has a Spin the Pointer game that uses a spinner like the one illustrated in the figure below.
A donation of $2 is requried to play the game. For each $2 donation, a player spins the pointer once and receives the amount of money indicated in the sector where the pointer lands on the wheel. The spinner has an equal probability of landing in each of the 10 sectors.
(a) Let X represent the net contribution to the charity when one person plays the game once. Complete the table for the probability distribution of X.
[pic]
(b) What is the expected value of the net contribution to the charity for one play of the game?
14. Two office assistants a Penny Lane High School are responsible for getting the daily tardy list to the appropriate principals by 3:00 p.m. daily. Rudy works on the lists 30% of the days and Fawn works on the tardy lists 70% of the days. Rudy fails to get the lists to the correct principals in time 10% of the time (which would make him tardy with the tardy lists). Fawn, not much better, manages to get the tardy lists to the correct principals 92% of the time. Let’s say you are Principal Sac and the tardy list is late. What is the probability that today Rudy is responsible for the list?
15. (2015 #3) A shopping mall has three automated teller machines (ATMs). Because the machines receive heavy use. They sometimes stop working and need to be repaired. Let the random variable X represent the number of ATMs that are working when the mall opens on a randomly selected day. The table shows the probability distribution of X.
[pic]
(a) What is the probability that at least one ATM is working when the mall opens?
(b) What is the expected value of the number of ATMs that are working when the mall opens?
(c) What is the probability that all three ATMs are working when the mall opens, given that at least one ATM is working?
(d) Given that at least one ATM is working when the mall opens, would the expected value of the number of ATMs that are working be less than, equal to, or greater than the expected value from part (b)? Explain.
16. Of the 60 obese teenages in a recent study, 15 had Type II diabetes, 20 had high blood pressure, and 10 had both high blood pressure and Type II diabetes. Suppose one of these 60 obese teenagers is randomly selected.
(a) Given that the teenager has Type II diabetes, what is the probability that he or she also has high blood pressure?
(b) If the obese teenager does NOT have high blood pressure, what is the probability that he or she also does not have Type II diabetes?
17. A machine that puts the center holes in blank CDs operates in such a way that the distribution of the diameter of the holes may be approximated by a normal distribution with a mean of 1.5 cm and a standard deviation of .1 cm. The specifications require the diameters of the holes to be between 1.4 and 1.6 cm. A CD not meeting the specifications is considered defective. (A center hole too small would not fit properly in a CD burner; a hole too large may cause the CD to slide during burning and ruin the quality of the music.) What proportion of CDs produced by this machine are defective due to an improperly sized center hole?
18. (2013 #3) Each full carton of Grade A eggs consists of 1 randomly selected empty cardboard container and 12 randomly selected eggs. The weights of such full cartons are approximately normally distributed with a mean of 840 grams and a standard deviation of 7.9 grams.
(a) What is the probability that a randomly selected full carton of Grade A eggs will weigh more than 850 grams?
(b) The weights of the empty cardboard containers have a mean of 20 grams and a standard deviation of 1.7 grams. It is reasonable to assume independence between the weights of the empty cardboard containers and the weights of the eggs. It is also reasonable to assume independence among the weights of the 12 eggs that are randomly selected for a full carton.
Let the random variable X be the weight of a single randomly selected Grade A egg.
i) What is the mean of X?
ii) What is the standard deviation of X?
19. The AP Statistics exam includes 40 multiple choice questions, each with 5 answer choices. Suppose you believe you have forgotten everything and must guess (randomly choose one of the five answer choices) on every question. Let x represent the number of correct responses on the test.
(a) What kind of probability distribution does x have? Explain.
(b) What is your expected score on the exam?
(c) Compute the variance and standard deviation of x?
(d) What is the probability that you will get exactly 25 questions correct?
(e) Overall, you believe you can do really well on the free response section. You did not study for the multiple choice section because you figured out that you would probably only need 16 questions correct to earn college credit in the course. What is the probability that you correctly answer at least 16 problems?
20. A plane used to fly tourists in and out of the rain forest contains seating for 16 passengers. The total weight limit for the passengers is 2500 pounds. Assume the average weight of tourists is 150 pounds, the standard deviation 27 pounds, and that the distribution of tourist weights is approximately normal. If the weight limit is exceeded, the plane has difficulty taking off safely. (We’re basically talking about crashing into very tall trees here. Not usually considered a vacation highlight!) If a random sample of 16 tourists has booked a flight, what is the chance that the weight limit will be exceeded?
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CJaNote: Adding/subtracting a constant only affects the mean. Multiplying/dividing affects both mean and variance.
Remember: To get standard deviation, take the square root of the variance.
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