# Prob/Stats Review

(17) Find the approximate percentile ranks of the following weights.

A. 57 pounds

b. 62 pounds

c. 64pounds

d. 59 pounds

Weight (pounds) Frequency

52.5-55.5 9

55.5-58.5 12

58.5-61.5 17

61.5-64.5 22

64.5-67.5 15 75

(19) Find the approximate scores that correspond to these percentiles.

a.15th

b. 29th

c. 43rd

d. 65th

e. 80th

Score Frequency

196.5-217.5 5

217.5-238.5 17

238.5-259.5 22

259.5-280.5 48

280.5-301.5 22

301.5-322.5 6

1 - Use your own paper as an answer sheet. Show all mathematical calculations. No partial credit will be given if work is not shown. A calculator will be required to complete the exam.

True or False (1 pt. Each)

3. Ordinal level of measurement consists of ranking data and precise differences between units of measure exist.

10. Dorothy asked 30 mothers how many children they had. The number of children is an example of a continuous variable.

Solving Problems

1. A questionnaire on housing arrangements showed the following information obtained from 30 respondents. Construct a frequency distribution and a pareto chart for the following data. H = house A = Apartment M = mobile home C = condominium

H C H M C A C A M C

M C A M A C C M C C

H A H H M C A M A C

2. When 36 randomly selected customers left a convenience store, each was asked the number of items he or she purchased. Construct a frequency distribution, histogram, and frequency polygon for the data.

2 9 4 3 6 6 6 2 8 6 5 5

7 5 3 8 6 6 6 2 3 2 4 4

6 9 9 8 9 9 4 2 1 7 4 4

3. The temperature of Singing River was recorded at 6 P.M. each day for the month of April. Construct a histogram, and ogive using relative frequencies for the data. Use six classes.

54 53 48 51 49 49 50 53 58 48 42 50

54 53 54 57 56 55 47 48 51 57 55 42

46 43 44 43 47 46

4 The following data represents the expenses of Chemistry Lab, Inc. for research and development for the years indicated. Each number represents millions of dollars. Draw a time series graph to represent the data. Be sure to use the same scale.

Year Amount

1965 $ 8,937

1971 9,388

1978 11, 271

1983 13, 271

1988 19, 203

5. In a study of 100 working teenagers, ages 16-18 the following numbers had given the major reasons for working as shown here. Construct a pie graph for the data.

Reason Number

To buy a car 45

To pay for education 39

For spending money 12

Other 4

6 The following temperatures were recorded in Nevada for a week in April.

87 85 80 78 83 86 90

Find each of the following: Must use formulas and show work.

a. mean

b. median

c. mode

d. midrange

e. range

f. variance- must use formula and show work

g. standard deviation

7. The outputs in volts of ten 9-volt batteries after 6 hours of use are shown. Be sure to use grouped data formulas.

Volts Frequency

3 1

4 3

5 4

6 1

7 1

Find each of the following: Must use formulas and show work

a) mean

b) median

c) mode

d) midrange

e) range

f) variance

g) standard deviation

h) the z-score for a 6 volt battery

8. Shown is a frequency distribution for the number of inches of rain received in one year in 25 selected cities in the United States.

Number of inches Frequency

5.5-20.5 2

20.5-35.5 3

35.5-50.5 8

50.5-65.5 6

65.5-80.5 3

80.5-95.5 3

Find each of the following: Must use formulas and show work.

a) mean

b) median

c) modal class

d) variance

e) standard deviation

9. In a survey of third grade students, the following distribution was obtained for the number of "best friends" each had. Find the average number of "best friends" for the class. Use the weighted mean.

Number of students Number of best friends

8 1

6 2

5 3

3 0

10. A sample of the labor costs per hour to assemble a certain product has a mean of $2.60 and a standard deviation of $0.15. Using Chebyshev's Theorem, find the values in which at least 88.89% of the data will lie.

11. The number of credits in business courses eight job applicants is shown:

9 12 15 27 33 45 63 72

a) Find the percentile rank for a value of 45

b) What value corresponds to the 40th percentile?

c) Construct a box and whisker plot and comment on the nature of the distribution.

12. The number of visitors to the Historic Museum for 27 randomly selected hours is shown below. Construct a stem and leaf plot for the data.

15 53 48 19 38 47 86 63 22

98 79 38 53 62 89 67 39 76

26 41 28 35 54 88 76 68 31

#2 -

1. A bowl contains 1 red, 1 black, and 1 yellow jellybean.

a) Two jellybeans are selected in succession, without the first jellybean being replaced. Draw a tree diagram and represent all possible combinations of jellybeans that can be selected.

b) Two jellybeans are selected in succession, with the first jellybean being replaced before the second is drawn. Draw a tree diagram and represent all possible combinations of jellybeans that can be selected.

2. How many seven digit telephone numbers are possible if digits can be repeated?

3. A doctor has five patients to visit in the next hour. How many different ways can he accomplish his rounds?

4. Using the numbers 0, 2, 4, 6, and 8, a salesman wants to construct a three-digit number. How many different numbers can be constructed?

5. A license plate must have 4 letters. The first letter must be N, the second and third letters can be anything, and the fourth letter must be a letter from "W, X, Y, or Z". How many different license plates are possible?

6. How many different ways can a moviegoer select three movies from a possible choice of nine movies?

7. A coin is tossed 9 times. How many different outcomes are there?

8. A new car buyer had the choice of 3 body styles, 2 engines and 7 colors. How many different ways are there to select a car?

9. How many different 3-letter permutations can be formed from the letters in the word "count".

10. A committee of 6 people must be selected from 9 men and 8 women. How many ways can this be done if there must be 3 of each gender on the committee?

11. How many ways can 5 books be arranged on a shelf, if they can be selected from 11 books?

12. When a card is drawn from a deck, find the probability of getting

a) diamond

b) a 5 or a heart

c) a 5 and a heart

d) a king

e) a red

13. In a recent survey 8 people watched ABC, 7 people watched NBC, and 12 people watched CBS. If a person is picked at random, what is the probability that the person watched NBC or CBS?

14. The probability that a person will get a cold this year is 0.7. If 4 people are selected at random, what is the probability that all 4 will get a cold this year?

15. Suppose you hold 30 tickets in a drawing for a color television. There were 500 tickets sold in the drawing. What is the probability that you win the color TV?

16. One card is drawn from an ordinary deck of playing cards. What is the probability of getting a card between 3 and 9?

17. When two dice are rolled, find the probability of getting

a) a sum of 4 or 11

b) a sum greater than 8

c) a sum less than 4 or greater than 9

18. If 85% of all people have brown eyes and six people are selected at random, find the probability that at least one of them has brown eyes?

19. In a College Algebra class there are 18 freshmen and 17 sophomores. Eight of the freshmen are females and 13 of the sophomores are males. If a student is selected at random, what is the probability of selecting a female?

20. Three cards are drawn from a deck without replacement. What is the probability of selecting a king, a nine, and a ten?

21. In a recent study, the following data were obtained in response to the question, "Do you favor the proposal of the school's combining the elementary and middle school students in one building?"

Yes No No opinion

Males 75 89 10

Females 105 56 6

If a person is selected at random, find these probabilities.

1. The person has no opinion

2. The person is a male or is against the issue.

3. The person is a female, given that the person opposes the issue.

22. The probability that a person has a VCR, given that he or she has a color T.V. is 0.45. The probability that a person has a color T.V. is 0.90. What is the probability that a person has a VCR and a color T.V.?

23. The number of cartoons watched by Mrs. Kelly's first grade class on Saturday morning is shown below. Find the mean, variance and standard deviation of the distribution.

X P(X)

0 0.15

1 0.20

2 0.30

3 0.10

4 0.20

5 0.05

24. If a gambler rolls two dice and gets a sum of 9, he wins $10, and if he gets a sum of four, he wins $20. The cost of the game is $5. What is the expectation of this game?

25. If 23% of all commuters ride to work in car pools, find the probability that if eight workers are selected, five will ride in car pools.

26. If 60% of all women are employed outside the home, find the probability that in a sample of 20 people:

a) Exactly 15 are employed

b) At least 10 are employed

c) At most 5 are retired

27. A company gave a drug test to its 300 employees. If the test is effective 98% of the time, find the mean and standard deviation for the number of correct drug tests.

28. 10% of the cameras sold in America are made in Europe, 70% are made in the U.S., and 20% are made in Japan. If 6 cameras are picked at random, what it the probability that 1 was made in Europe, 3 in the U.S. and 2 in Japan?

29. If 4% of the population catch a cetrain disease, find the3 probability that in a group of 200 people, 3 will get the disease. This approximates a Poisson.

30. A door-to-door salesman averages 3 sales a day. Find the probability of getting 6 sales in a day if this approximates a Poisson.

31. A bag has 10 red and 20 blue marbles. If you pull out 5 without replacement, find the probability of getting 3 red and 2 blue marbles.

32. Find the area under the normal distribution curve

a) to the right of 2.45 and to the left of - 2.08

33. Find the probabilities for each, using the standard normal distribution

a) P ( z < - 2.34 )

34. Find the z value to the right of the mean so that 45% of the area under the curve lies to the left of it.

35. A light bulb is advertised as lasting 1000 hours with a standard deviation of 150 hours. Find the probability of buying a light bulb and having it last:

a) less than 1250 hours

b) between 901 and 1000 hours

c) more than 901 hours

36. A certain diet claims to lower cholesterol by an average of 60 points with a standard deviation of 7. Find the probability that the mean amount lowered will be more than 75 points.

37. Mensa is an organization that accepts members who score in the top 2% of the population on standard intelligence tests. If the mean IQ is 100, with a standard deviation of 15, what should the minimum IQ score be to be accepted by Mensa?

38. The average hourly wage of fast-food workers employed by a nationwide chain is $5.55. The standard deviation is $1.15. If a sample of 50 workers is selected, find the probability that the mean of the sample will be between $5.25 and $5.90.

39. A librarian knows that 14% of the borrowers will not return the books by their due dates. If a sample of 75 people that have checked out books from the library is tested, find the probability that more than 15 will not return the books on time.

***See attached file***

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#### Solution Preview

(17) Find the approximate percentile ranks of the following weights.

a. 57 pounds

b. 62 pounds

c. 64pounds

d. 59 pounds

Weight ( pounds) Frequency

52.5-55.5 9

55.5-58.5 12

58.5-61.5 17

61.5-64.5 22

64.5-67.5 15 75

Something in the xth percentile is greater than or equal to x% of the data. The relationship between the rank of something in the list of data and the percentile can be calculated by

R = (P/100)(N + 1)

where R is the rank, P is the percentile, and N is the sample size. [Note: there is more than one definition of percentile. Some people would calculate using "N" instead of "N + 1". Since we're approximating anyway, it doesn't really matter to us.]

a. 57 pounds. This weight is halfway into the second interval. Therefore, its rank can be approximated as 15 (when sorting the data from smallest to largest, the number 57 would be the 15th in the list). Using the formula, you can see that 57 lbs is at approximately the 20th percentile.

15 = (P/100)(76)

19.74 = P

Or, if you used N instead of N + 1, you'd get P = 20.

b. 62 pounds. This weight is about 1/6 into the 4th interval. Its rank can be approximated as 9 + 12 + 17 + 22/6 = 41.667 ≈ 42. This weight is at the 56th percentile.

c. 64 pounds. This weight is about 5/6 into the 4th interval. Its rank can be approximated as 9 + 12 + 17 + 22*5/6 = 56.333 ≈ 56. This weight is at the 75th percentile.

d. 59 pounds. This weight is about 1/6 into the 3rd interval. Its rank can be approximated as 9 + 12 + 17/6 = 23.833 ≈ 24. This weight is at the 32nd percentile.

(19) Find the approximate scores that correspond to these percentiles.

a.15th

b. 29th

c. 43rd

d. 65th

e. 80th

Score Frequency

196.5-217.5 5

217.5-238.5 17

238.5-259.5 22

259.5-280.5 48

280.5-301.5 22

301.5-322.5 6

This is similar to the last problem. There are 120 total scores. We need to find the scores that are 15%, 29%, etc. of the way down the list.

a. 15th The score 15% of the way down the list would have a rank of 120*0.15 = 18. This score would be the 13th score in the second interval. The second interval has a width of 21 (238.5 - 217.5 = 21). The 13th score out of the 17 values in that interval would be about ¾ of the way into the list (13/17 = 0.76). This would be approximately equal to 233.5.

b. 29th The score 29% of the way down the list would have a rank of 34. This score would be the 13th score in the third interval. The 13th score out of the 22 values would be approximately equal to 250.

c. 43rd The score 43% of the way down the list would have a rank of 51. This score would be the 7th score in the fourth interval. The 7th score out of the 48 values would be approximately equal to 263.

d. 65th The score 65% of the way down the list would have a rank of 78. This score would be the 14th score in the fourth interval. The 14th score out of the 48 values would be approximately equal to 266.

e. 80th The score 80% of the way down the list would have a rank of 96. This score would be the 4th score in the fifth interval. The 4th score out of the 22 values would be approximately equal to 284.

TRUE OR FALSE

3. Ordinal level of measurement consists of ranking data and precise differences between units of measure exist. FALSE. In ordinal measurement, the data are rankings, but only interval or ratio scales have "precise differences between units of measure."

10. Dorothy asked 30 mothers how many children they had. The number of children is an example of a continuous variable. FALSE. This is a discrete variable, because you can only have 0, 1, 2, 3, etc. children. You can't have any non-integer number of children.

SOLVING PROBLEMS

1. (5pts.) A questionnaire on housing arrangements showed the following information obtained from 30 respondents. Construct a frequency distribution and a pareto chart for the following data. H = house A = Apartment M = mobile home C = condominium

H C H M C A C A M C

M C A M A C C M C C

H A H H M C A M A C

type frequency

H 5

A 7

M 7

C 11

2. (5 pts.) When 36 randomly selected customers left a convenience store, each was asked the number of items he or she purchased. Construct a frequency distribution, histogram, and frequency polygon for the data.

2 9 4 3 6 6 6 2 8 6 5 5

7 5 3 8 6 6 6 2 3 2 4 4

6 9 9 8 9 9 4 2 1 7 4 4

From top to bottom, here are the frequency distribution, histogram, and frequency polygon. Let me know if you need help in constructing any of these.

Number Frequency

1 1

2 5

3 3

4 6

5 3

6 8

7 2

8 3

9 5

3. (5pts.) The temperature of Singing River was recorded at 6 P.M. each day for the month of April. Construct a histogram, and ogive using relative frequencies for the data. Use six classes.

54 53 48 51 49 49 50 53 58 48 42 50

54 53 54 57 56 55 47 48 51 57 55 42

46 43 44 43 47 46

The range is 16. If we have 6 classes, each class will have an interval of 3.

Interval Frequency Cum. Freq.

41 - 43 4 4

44 - 46 3 7

47 - 49 7 14

50 - 52 4 18

53 - 55 8 26

56 - 58 4 30

Histogram:

Ogive:

4 (5pts.) The following data represents the expenses of Chemistry Lab, Inc. for research and development for the years indicated. Each number represents millions of dollars. Draw a time series graph to represent the data. Be sure to use the same scale.

Year Amount

1965 $ 8,937

1971 9,388

1978 11, 271

1983 13, 271

1988 19, 203

5. (5pts.) In a study of 100 working teenagers, ages 16-18 the following numbers had given the major reasons for working as shown here. Construct a pie graph for the data.

Reason Number

To buy a car 45

To pay for education 39

For spending money 12

Other 4

I did this in Excel. To do this by hand, convert the numbers into percentages (i.e. 45% of teenagers worked to buy a car), and divide a circle into the appropriate sized percentages.

6 (10pts.) The following temperatures were recorded in Nevada for a week in April.

87 85 80 78 83 86 90

Find each of the following: Must use formulas and show work.

a. mean: The mean is the average of all the numbers. You calculate it by adding all the numbers together and dividing by the number of observations.

mean = 589/7 = 84.1

b. median: The median is the middle number. You find it by sorting all the numbers from smallest to largest, then taking the number in the middle of the list.

median = 85

c. mode: The mode is the number that appears the most often. There is no mode here, since all the numbers only appear once.

d. midrange: The midrange is the mean of the smallest and largest numbers.

midrange = (78 + 90)/2 = 84

e. range: The range is the maximum minus the minimum.

range = 90 - 78 = 12

f. variance- must use formula and ...

#### Solution Summary

This problem set and the solutions/explanations can be used to study for an introductory probability and statistics course.