Download Review for first semester exam

Statistics AP Review for First Semester Exam
Exam covers Chapters 1 - 16
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The final exam will be free response and multiple-choice.
To prepare for the exam I am giving you a review and a list of topics to review. I would also encourage you to review
all quizzes that we have had. If you have an AP study book you could also review the part that this test covers.
I would strongly encourage you to hold on to this review so that you can use it in preparation for the AP testing May.
This reviews half of our course so I feel it should be a good resource in the spring.
Review Problems
1. Suppose the average score on a national test is 500 with a standard deviation of 100. If each value is increased by 25,
what are the new mean and standard deviation values?
2. Suppose the average score on a national test is 500 with a standard deviation of 100. If each score is increased by 25%,
what are the new mean and standard deviation?
3. Which of the following statements are true?
I. If the right and left sides of a histogram are mirror images of each other, the distribution is symmetric.
II. A distribution spread far to the right side is said to be skewed to the right.
III. If a distribution is skewed to the right, its mean is greater than its median.
4. According to The New York Times (April 2, 1993, page A1), the average monthly rate for basic television cable service
has increased as follows:
Year :
1986
1987 1988 1989 1990 1991 1992
Rate($): 11.00 13.20 13.90 15.20 16.80 18.00 20.00
(a) Find the equation of the best-fitting straight line.
(b) Interpret the slope.
(c) Predict the average monthly rate in 1993.
(d) Predict what year the rate will reach $50.00.
(e) What is the correlation coefficient? What is the coefficient of determination?
(f) Explain r-squared in the context of the problem.
(g) Draw a residual plot of this data. Explain what this tells you.
5. Some researchers believe that too much iron in the blood can raise the level of cholesterol. Making periodic blood
donations can lower the iron level in the blood. A study is performed by randomly selecting half of a group of volunteers
to give periodic blood donations while the other half does not give blood donations. Is this an experiment or an
observational study?
6. The probability that a student will receive a state grant is 1/3, while the probability that she will be awarded a federal
grant is ½. If whether or not she receives one grant is not influenced by whether or not she receives the other, what is the
probability of her receiving both grants?
7. Suppose a reputed psychic in an extrasensory perception (ESP) experiment has called heads or tails correctly on ten
successive tosses of a coin. What is the probability that guessing would have yielded this perfect score?
8. A videocassette recorder (VCR) manufacturer receives 70% of his parts from factory F1 and the rest from factory F2.
Suppose that 3% of the output from F1 is defective, while only 2% of the output from F2 is defective. What is the
probability a received part is defective?
9. A manager notes that there is a 0.125 probability that any employee will arrive late for work. What is the probability
that exactly one person in a six-person department will arrive late?
10. A manufacturer has the following quality control check at the end of a production line: If at least eight of ten randomly
picked articles meet all specifications, the whole shipment is approved. If, in reality, 85% of a particular shipment meets
all specifications, what is the probability that the shipment will make it through the control check?
11. Using number 10 above, what is the probability that a shipment in which only 70% of the articles meet specifications
will make it through the control check?
12. Joe DiMaggio had a career batting average of .325. What is the probability that he would get at least one hit in five
official times at bat?
13. A grocery store manager notes that 35% of customers who buy a particular product make use of a store coupon to
receive a discount. If seven people purchase the product, what is the probability that fewer than four will use a coupon?
14. Concessionaires know that attendance at a football stadium will be 60,000 on a clear day, 45,000 if there is light snow,
and 15,000 if there is heavy snow. Furthermore, the probability of clear skies, light snow, or heavy snow on any particular
day is ½, 1/3, and 1/6., respectively. What average attendance should be expected for the season?
15. In a lottery, 10,000 tickets are sold at $1 each with a prize of $7500 for one winner. What is the average result for each
bettor?
16. A manager must choose among three options. Option A has a 10% chance of resulting in a $250,000 gain but
otherwise will result in a $10,000 loss. Option B has a 50% chance of gaining $40,000 and 50% chance of losing $2000.
Finally, option C has a 5% chance of gaining $800,000 but otherwise will result in a loss of $20,000. Which option should
the manager choose?
17. Of the automobiles produced at a particular plant, 40% had a certain defect. Suppose a company purchases five of
these cars. What is the expected value of the number of cars with defects?
18. The life expectancy of a particular brand of lightbulb is normally distributed with mean of 1500 hours and a standard
deviation of 75 hours.
(a) What is the probability that a lightbulb will last less than 1410 hours?
(b) What is the probability that a lightbulb will last between 1416 and 1450 hours?
(c) What is the probability that a lightbulb will last between 1563 and 1648 hours?
19. A packing machine is set to fill a cardboard box with a mean average of 16.1 ounces of cereal. Suppose the amounts
per box form a normal distribution with a standard deviation of 0.04 ounces.
(a) Ten percent of the boxes will contain more than what number of ounces?
(b) Eighty percent of the boxes will contain more than what number of ounces?
(c) What percentage of the boxes will end up with at least 1 pound of cereal?
(d) The middle 90% of the boxes will be between what two weights?
20. One- thousand students at a city high school were classified according to both GPA and whether or not they
consistently skipped school.
GPA
<2.0
2.0-3.0
>3.0
Many skipped school
80
25
5
Few skipped school
175
450
265
(a) What is the probability that a student has a GPA between 2.0 and 3.0?
(b) What is the probability that a student has a GPA under 2.0 and has skipped many classes?
(c)What is the probability that a student has a GPA under 2.0 or has skipped many classes?
(d) What is the probability that a student has a GPA under 2.0 given that he has skipped many classes?
(e) Are events “GPA between 2.0 and 3.0” and “skipped many classes” independent events? Explain.
21. When describing a graphical display, if the shape is approximately symmetric, you should use _________ as a
measure of center and _______ as a measure of spread.
If the shape is skewed, you should use _______ as a measure of center and ________ as a measure of spread.
22. Mathematically speaking, casinos and life insurance companies make a profit because:
(A) Of their understanding of sampling error and source of bias.
(B) Of their use of well-designed, well-conducted surveys and experiments.
(C) Of their use of simulation of probability distributions.
(D) Of the law of large numbers.
23. A club sells raffle tickets and there are 10 prizes at $25 and one prize of $100. If 200 tickets are sold, what do you
expect to win if you bought one ticket? If the ticket cost $5, do you expect to lose or gain? Explain
24. The probability model below describes the number of repair calls than an appliance repair shop may receive during an
hour.
Repair Calls
0
1
2
3
Probability
0.1
0.3
0.4
0.2
(a) How many calls should the shop expect per hour?
(b) What is the standard deviation?
(c)Find the mean and standard deviation of the number of repair calls the appliance shop should expect during an 8 - hour
day.
25. Insurance companies compute expected values so that they can set their rates at profitable but competitive levels. A
64-year old man obtains a $10,000 one year life insurance policy at a cost of $600. Based on past mortality experience,
the insurance company estimates that there is a 0.963 chance that this man will live for at least one year. How much can
the insurance company expect to earn on this policy?
26. A car owner cannot decide whether to take out a $250 deductible which will cost him $90 per year. Records show that for his
community the average cost of repair is $900. Records also show that 10% of the drivers have an accident during the year.
(a) Find the expected cost of the car owner if he buys the policy.
(b) Find the expected cost of the car owner if he does not buy the policy.
(c) Assuming the car owner would not be financially handicapped if he were to pay out the $900 or more for repairs would you
suggest that he buy the policy?
27. A company has 11 mathematicians on its staff, of which 3 are women. The president of the company is concerned about the
number of women mathematicians. The president learns that about 30% of the mathematicians in the U.S. are women, and asks you to
investigate whether or not the number of women mathematicians in the company is consistent with the national pool. The president
knows very little about statistics. You decide the answer the president’s question with a simulation.
(a) Write your assumptions
(b) Describe how you will use the table to model this situation and describe how you will find one trial for this simulation
(c) Using the table provided, write above the table and communicate your results from finding three trials.
1 9 2 2 3
9 5 0 3 4
0 5 7 5 6
2 8 7 1 3
9 6 4 0 9
1 2 5 3 1
4 2 5 4 4
8 2 8 5 3
7 3 6 7 6
4 7 1 5 0
9 9 4 0 0
0 1 9 2 7
8 6 6 4 9
2 4 7 5 5
(d) Using the calculator find the results of 20 more trials. Construct a probability distribution for this situation. Be sure to explain your
random variable. Record your results in a frequency distribution.
(e) Using your results what is the probability that no more than 3 of the 11 mathematicians would be women?
Unit I: Chapters 1 - 5 (35% of the final exam content)
1. Review definitions of populations and samples.
2. Review various graphical displays. (Review when to use which type, how to use the calculator to graph box plots and histograms,
how to all of them without the calculator, which type is used for discrete data and which are used for continuous data.)
3. Review terms on pages 71-77.
4. When describing a graphical display for univariate data, you must “CUSS” center, unusual, shape, and spread.
5. Know the symbols for population parameters and sample statistics (i.e. the mean, standard deviation, and variance of populations
and samples.
6. Know the rule for finding an outlier for univariate data.
7. Know the probabilities for the Empirical Rule.
8. Know how to apply the Normal model – draw the curve, write probability, use the tables.
Extra Problems: Unit 1 Review- Textbook pages 138-146
Unit 1 Practice Exam- Textbook pages 147-149
Unit II: Chapters 6-9 (11% of the final exam content)
1. Know how to find a regression equation.
2. Know how to explain the correlation coefficient in context of the problem.
3. Review the properties of “r.”
4. Review the information on correlation and causation.
5. Review the regression equation formula on formula chart. Review what the value of “b” represents.
6. Review how to find the coefficient of determination and what it represents.
7. How does a statistician determine whether a value is an outlier in a scatter plot? How do they determine an influential point?
8. Know how to interpret the slope and the coefficient of determination in context of a problem.
9. Review how to find a residual (both by hand and with a calculator)
10. Review what a residual plot tells you about the data.
Extra Problems: Unit II Review- Textbook pages 255-264
Unit II Practice Exam- Textbook pages 264-266
Unit III: Chapters 10-12 (21% of the final exam content)
1. Review observational study vs. experiment.
2. Review the various methods of sampling (simple random sampling, stratified sampling, and systematic sampling.)
3. Recognize various types of biases.
4. Review purpose of experiments. Review key concepts of experiments.
5. Review simulations. – look over the simulation packet
Extra Problems: Unit III Review- Textbook pages 331-336
Unit III Practice Exam- Textbook pages 336-342
Unit IV: Chapters 13-16 (33% of the final exam content)
1. Review the Law of Large Numbers.
2. Review Laws of Probability
3. Review tree diagrams and tables
4. Compare the difference between a discrete random variable and a continuous random variable.
5. Know how transformations on the random variable affect the mean, variance, and standard deviation.
6. Know how to find the mean, variance, and the standard deviation sum/difference of random variables.
7. Review formulas for expected values, variance, and standard deviation for random variables, binomial, and geometric distributions.
8. Review how to use the calculator here.
9. Know how to use geometric and binomial models.
Extra Problems: Unit IV Review- Textbook pages 434-439
Unit IV Practice Exam- Textbook pages 439-444
The final exam will have two parts:
1) Multiple Choice (67% of exam grade)
2) Free Response (33% of exam grade)