statistics

Practice Problems for Statistics Final Exam

Dr. Rispoli

Chapters 2 Descriptive Statistics

1. Thirty automobiles were tested for fuel efficiency in mpg. The following frequency distribution was obtained.

Class Boundaries Frequency

7.5-12.5 3

12.5-17.5 5

17.5-22.5 15

22.5-27.5 5

27.5-32.5 2

(a) Construct a histogram, frequency polygon and ogive for the data.

(b) Find the mean for this data.

2. The number of credits in business courses that twelve students took is shown below.

5, 6, 9, 12, 15, 21, 21, 27, 33, 45, 63, 72

(a) Find the percentile rank of 33.

(b) Find a value that that corresponds to the 40th percentile.

(c) Construct a box-plot labeled with the 5 number summary.

3. The data shown (in millions of dollars) are the values of the 30 NFL franchises in the year 2008.

170 191 171 235 173 187 181 191

200 218 243 200 182 320 184 239

186 199 186 210 209 240 204 193

211 186 197 204 188 242

(a) Construct a frequency distribution for the data using 8 classes.

(b) Construct a histogram for the data.

(c) Construct a frequency polygon and an ogive for the data.

4. The following data are based on a survey of why people travel. Construct a Pareto chart for this data.

Purpose Number

Personal Business 146

Visit friends or relatives 350

Work-related 325

Leisure 299

Attend School 150

Evaluate Airlines 14

5. a) Find the five-number summary for the following data below.

b) Construct a box-plot.

c) Find the IQR and use it to determine if there are any outliers.

30 28 33 29 37 39 57 27 16 25

35 37 37 38 34 25 21 3 34 35

Chapter 3 Basic Probability

1. A study of graduates average grades and degrees showed the following results.

Grade

Degree C B A

B.S. 5 8 15

B.A. 7 12 8

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

(a) The graduate has an average grade of an A or B.

(b) The graduate has a B.A. degree and an average of a B.

(c) The graduate has a B.A. degree, given that he or she has an A average.

(d) Given that the graduate has a B.A. degree, the graduate has an C average.

2. The probability that a person owns a car is .80, that a person owns a boat is .30, and that a person owns both is .12. Find the probability that a person owns either a car or a boat, but not both.

3. A roulette wheel has 38 spaces numbered 1 through 36, 0, and 00. Find the probability of getting:

a) a number less than 15 not counting 0 and 00

b) a number that is a multiple of 3 or 5 not counting 0 and 00

c) an odd number less than 15 not counting 0 and 00

4. A number of students were grouped according to their reading ability and education. The table shows the results.

Education Low Average High

Graduated high school 6 18 43

Did not graduate 27 16 7

If a student is selected at random, find the probability that:

a) The student graduated high school.

b) The student has a low reading ability, given that the student is a high school graduate. c) The student does not have a high level of reading ability.

d) The student did not graduate high school, given that the student has an average reading ability.

5. Five cards are drawn from an ordinary deck without replacement. Find the probability of getting:

a) All red cards

b) All diamonds

c) All aces

d) A royal flush

6. The number of endangered species for several groups are listed below.

Mammals Birds Reptiles Amphibians

United States 65 80 25 15

Foreign 250 175 75 10

If one endangered species is selected at random, find the probability that it is:

a) Found in the United States and is a bird.

b) Foreign or a mammal.

c) A bird, given that it is found in the United States.

d) Foreign, given that it is a reptile.

7. When 3 dice are rolled find the probability of getting a sum of 7.

8. At a recent graduation at a naval flight school, 18 Marines, 10 members of the Navy and 3 members of the Coast Guard got their wings. If we chose three pilots at random to feature on a training brochure, find the probability that there will be:

a) 1 member from each branch of the service

b) 0 members of the Navy

c) 3 Marines

9. Eighty-one percent of US households have DVD players. Choose 6 households at random. What is the probability that at least 1 does not have a DVD player?

Chapter 4 Discrete Probability Distributions

1. At a certain store the number of arrivals per minute during the day has the following distribution:

Number of arrivals, x 0 1 2 3 4 5 6

Probability, P(x) .10 .20 .25 .18 .14 .08 .05

Find the mean and standard deviation of the distribution.

2. A box contains 5 pennies, 13 dimes, 10 quarters and 2 half-dollars. A coin is selected from the box at random. Construct the probability distribution by filling in the table below and draw a graph for the data.

Type of coin selected , x Penny Dime Quarter Half-Dollar

Probability, P(x)

3. If 80% of all people over 60 are retired, find the following probabilities for a sample of 20 people. Use the binomial distribution

(a) Exactly 15 are retired.

(b) At least 15 are retired.

(c) At most 15 are retired.

4. If 40% of all commuters ride to work in carpools, find the probability that if 8 workers are selected:

a) exactly 5 will ride in carpools;

b) at most will ride in carpools;

c) at least 5 will ride in carpools.

5. If 80% of all job applicants are able to pass a computer literacy test, find the mean, variance and standard deviation of people who pass the examination in a sample of 150. Show all calculations.

6. Let X be a binomial random variable with n = 12 and p = 0.3. Find the following:

a) P(X = 7)

b) P(X < 4)

c) P(X > 8)

d) P(4 < X < 10)

7. Ray Allen, perhaps the best 3-point shooter over the last 10 years in the NBA, has a chance to shoot 4 free throws. He was fouled shooting a 3 point shot, and he gets to take a fourth shot due to a technical foul. Suppose that the probability that he makes a free throw is .9, and his free throws are independent of each other. Let X be the random variable that gives the number of free throws made in 4 attempts.

a) Give the possible values for X.

b) Obtain the probability distribution for x.

Chapter 5 The Normal Distribution

For all problems be sure to use the following 3 steps:

(i) Sketch a bell curve and shade the area corresponding to the probability you want

(ii) Find the appropriate z-score(s)

(iii) Determine the appropriate probability

1. The number of gallons of Gatorade consumed by a football team during a game follows a normal distribution with mean 20. The standard deviation is 3. If a game is selected at random, find the probability that the number of gallons consumed will be:

(a) Greater than 18 gallons.

(b) Between 22 and 25 gallons.

(c) Less than 16 gallons.

2. The average diastolic blood pressure of a certain age group of people is 85 mmHg, and follows a normal distribution. The standard deviation is 6. If an individual is selected at random, find the probability that the individual's pressure will be:

(a) Greater than 90

(b) Between 90 and 95.

(c) Between 82 and 90.

3. An automobile dealer finds that the average price of a used car is $9,000. He decides to sell cars that appeal to the middle 70% of the market in terms of price. Find the minimum and maximum prices of the cars the dealer will sell. The standard deviation is $1,300 and the distribution of prices is normal.

4. The average weight of a group of young adults is 160 pounds. The standard deviation is 10. If a sample of 49 is taken from this group, find the probability that the sample mean for the 49 will be less than 155 pounds.

Chapter 6 Confidence Intervals and Sample Size

1. A study of 40 bowlers showed that their average score was 186. The standard deviation of the population is 6.

(a) Construct a 95% confidence interval for the mean score of all bowlers.

(b) Construct a 95% confidence interval for the mean score of all bowlers if a sample of size 100 is used instead of 40.

(c) Explain why one confidence interval is larger than the other.

2. A health care professional wishes to estimate the birth weights of infants. How large a sample must she select if she desires to be 90% confident that the true mean is within 6 ounces of the sample mean? The standard deviation is estimated to be 8 ounces.

3. A random sample of 78 students were interviewed and 59 said they would vote for Jennifer McNamara as student body president. Let p represent the proportion of all students who will vote for Jennifer. Find a 90% confidence interval for p.

4. As part of an Environmental Studies class project students measured the circumference of a random sample of 45 Blue Spruce trees near Brainard Lake, Colorado. The sample mean circumference was 29.8 inches. Assume that ( is known to be 7.2 inches. Find a 95% confidence interval for the population mean circumference of all Blue Spruce trees near this lake.

5. A random sample of 56 credit card holders showed that 41 regularly paid their credit card bills on time. Find a 95% confidence interval for the proportion of all people who regularly paid their credit card bill on time.

6. A researcher wants to know what percentage of males athletes wear contact lenses during performances. Let p represent the proportion of males athletes who wear contact lenses. If there is no preliminary estimate for p, how many male athletes should be included in a random sample to be 90% sure that a point estimate for p will be within a distance of 0.05 from p.

7. A sample of 25 novels has a standard deviation of 9 pages. Find the 95% confidence interval of the population standard deviation.

Chapter 7 Hypothesis Testing

Use the following 5 step method for all problems.

(i) State the hypotheses H0, H1, and identify the claim (notation is important, use population parameters)

(ii) Find the critical value(s) from the appropriate distribution, check conditions to determine the appropriate distribution

(iii) Compute the test value using the appropriate formula

(iv) Make a decision to reject or not reject the null hypothesis by comparing the test value to the critical value.

(v) Summarize the results in the context of the problem.

1. A manufacturer states that the average lifetime of its light bulbs is 3 years. The standard deviation is  = 8 months. A sample of fifty bulbs is taken and the average lifetime is found to be 34 months.

(a) Should the claim be rejected at the  = .05?

(b) Should the claim be rejected at the  = .10?

2. A travel agency says that it books an average of 42 people per trip to Atlantic City. A sample of 10 trips showed a mean of 48 people booked and a sample standard deviation of s = 8. At  = .01, test the claim.

3. Experts claim that 10% of murders are committed by women. Is there enough evidence at  = .05 to reject the claim if in a sample of 67 murders, 10 were committed by women?

4. A bank manager claims that the average loan to the bank's customers is $4800. The population standard deviation is $800. A sample of 25 customers had an average loan of $4235. At  = .10, does the evidence support the manager's claim?

5. How long does it take juniors to complete a standardized exam? The long-term average is 2.8 hours. We may assume that x has a normal distribution with  = 0.8. A random sample of 12 juniors gave a sample mean of 2.2 hours. Does this indicate that the population mean time is different from 2.8 Hours? Use 5% level of significance.

6. A manufacturer claims that the standard deviation of the drying time of a certain type of paint is 18 minutes. A sample of five test panels produced a standard deviation of 21 minutes. Test the claim at  = .05.

Chapter 9 Correlation and Regression

1. A study is done to see whether there is a relationship between a student’s GPA and the number of hours the student studies per week. The data is as follows.

Hours, x 3 12 9 15 5 7 16

GPA, y 2.1 3.5 3.0 4.0 1.7 3.2 3.7

(a) Draw a scatter plot.

(b) Compute the value of the correlation coefficient r.

(c) Test the significance of the correlation coefficient at  = .01

(d) Find the equation for the least squares line.

(e) Graph the least squares line on your scatter plot.

2. Does the weight of a vehicle affect gas mileage? The following random sample was collected where x = weight of a vehicle in hundreds of pounds and y = miles per gallon.

x 26 35 29 39 20

y 22.0 16.1 18.8 15.7 23.4

a) Create a scatter plot for this data.

b) Compute the coefficient of correlation.

c) Test the significance of the correlation coefficient at  = .05.

d) Find the equation for the least squares line.

e) If the weight of a vehicle is 32, what do you predict the gas mileage will be?

3. An emergency service wishes to see whether a relationship exists between the outside temperature and the number of emergency calls it receives for a 7-hour period. The data is given below.

Temperature x 68 74 82 88 93 99 101

Number of calls, y 7 4 8 10 11 9 13

a) Create a scatter plot for this data.

b) Compute the coefficient of correlation. You may use the following sums.

( x = 605 ( y = 62 ( xy = 5,535 ( x2 = 53,219 ( y2 = 600.

c) Test the significance of the correlation coefficient at  = .05.

d) Find the equation for the least squares line.

e) If the temperature is 85, what do you predict the number of calls will be?

f) What percent of the variation in y is captured by the model?