# EBook Problems

## Probability and Statistics EBook Practice Problems

The problems provided below may be useful for practicing the concepts, methods and analysis protocols, and for self-evaluation of learning of the materials presented in the EBook.

# II. Describing, Exploring, and Comparing Data

## Pictures of Data

1. Two random samples were taken to determine backpack load difference between seniors and freshmen, in pounds. The following are the summaries:

 Year Mean SD Median Min Max Range Count Freshmen 20.43 4.21 17.20 5.78 31.68 25.9 115 Senior 18.67 3.56 18.67 5.31 27.66 22.35 157

Which of the following plots would be the most useful in comparing the two sets of backpack weights?

A. Histograms

B. Dot Plots

C. Scatter Plots

D. Box Plots

## Measures of Central Tendency

1. Suppose that in a certain country, the average yearly income for 75% of the population is below average, what would you use as the measure of center and spread?

A. Mean and interquartile range

B. Mean and standard deviation

C. Median and interquartile range

D. Mean and standard deviation

## Measures of Variation

1. The number of flaws of an electroplated automobile grill is known to have the following probability distribution:

 X 0 1 2 3 P(X) 0.8 0.1 0.05 0.05

What would be the standard deviation of the sample means if we took 100 samples, each sample with 200 grills, and computed their sample means?

A. 0.6275

B. 0.0560

C. None of the Above

D. 0.89269

2. Suppose that in a certain country, the average yearly income for 75% of the population is below average, what would you use as the measure of center and spread?

A. Mean and interquartile range

B. Mean and standard deviation

C. Median and interquartile range

D. Mean and standard deviation

# III. Probability

## Fundamentals

1. In a large midwestern university with 30 different departments, the university is considering eliminating standardized scores from their admission requirements. The university wants to find out whether the students agree with this plan. They decide to randomly select 100 students from each department, send them a survey, and follow up with a phone call if they do not return the survey within a week. What kind of sampling plan did they use?

A. Stratified random sampling

B. Simple random sampling

C. Multi-stage sampling

D. Cluster sampling

## Rules for Computing Probabilities

1. A professor who teaches 500 students in an introductory psychology course reports that 250 of the students have taken at least one introductory statistics course, and the other 250 have not taken any statistics courses. 200 of the students were freshmen, and the other 300 students were not freshmen. Exactly 50 of the students were freshmen who had taken at least one introductory statistics course.

If you select one of these psychology students at random, what is the probability that the student is not a freshman and has never taken a statistics course?

A. 30%

B. 40%

C. 50%

D. 60%

E. 20%

2. A box contains 30 pens, where 5 are red, 14 are black, and 11 are blue. If you pick three pens from the box at random without replacement, what is the probability that these three pens will all be black?

A. 14/30 + 14/30 + 14/30

B. 14/30 + 13/29 + 12/28

C. 14/30 x 13/29 x 12/28

D. 1 - (14/30 x 13/29 x 12/28)

3. When three fair dice are simultaneously thrown, which of these three results is least likely to be obtained?

A. All three results are equally unlikely.

B. Two fives and a 3 in any order.

C. A 5, a 3 and a 6 in any order.

D. Three 5's.

4. Suppose that you take a three question "true/false" quiz for which you are completely unprepared. You have to guess the correct answer for each question. What is the probability of answering at least one question correctly?

A. 4/8

B. 5/8

C. 7/8

D. 1/8

E. 3/8

# IV. Probability Distributions

## Expectation(Mean) and Variance)

1. Ming’s Seafood Shop stocks live lobsters. Ming pays \$6.00 for each lobster and sells each one for \$12.00. The demand X for these lobsters in a given day has the following probability mass function.

 X 0 1 2 3 4 5 P(x) 0.05 0.15 0.3 0.2 0.2 0.1

What is the Expected Demand?

A. 13.5

B. 3.1

C. 2.65

D. 5.2

# V. Normal Probability Distribution

## The Standard Normal Distribution

1. Weight is a measure that tends to be normally distributed. Suppose the mean weight of all women at a large university is 135 pounds, with a standard deviation of 12 pounds. If you were to randomly sample 9 women at the university, there would be a 68% chance that the sample mean weight would be between:

A. 131 and 139 pounds.

B. 133 and 137 pounds.

C. 119 and 151 pounds

D. 125 and 145 pounds.

E. 123 and 147 pounds.

## Nonstandard Normal Distribution: Finding Probabilities

1. A researcher converts 100 lung capacity measurements to z-scores. The lung capacity measurements do not follow a normal distribution. What can we say about the standard deviation of the 100 z-scores?

A. It depends on the standard deviation of the raw scores

B. It equals 1

C. It equals 100

D. It must always be less than the standard deviation of the raw scores

E. It depends on the shape of the raw score distribution

# VI. Relations Between Distributions

==The Central Limit Theorem==A. A smaller population standard deviation 1. Which of the following would make the sampling distribution of the sample mean narrower? Check all answers that apply.

A. A smaller population standard deviation

B. A smaller sample size

C. A larger standard error

D. A larger sample size

E. A larger population standard deviation

# VII. Point and Interval Estimates

## Estimating a Population Mean: Large Samples

1. Two researchers are going to take a sample of data from the same population of physics students. Researcher A will select a random sample of students from among all students taking physics. Researcher B's sample will consist only of the students in her class. Both researchers will construct a 95% confidence interval for the mean score on the physics final exam using their own sample data. Which researcher's method has a 95% chance of capturing the true mean of the population of all students taking physics?

A. Research B

B. Researcher A

C. Both methods have a 95% chance of capturing the true mean

D. Neither

## Estimating a Population Proportion

1. A 1996 poll of 1,200 African American adults found that 708 think that the American dream has become impossible to achieve. The New Yorker magazine editors want to estimate the proportion of all African American adults who feel this way. Which of the following is an approximate 90% confidence interval for the proportion of all African American adults who feel this way?

A. (.56, .62)

B. (.57, .61)

C. Can't be calculated because the population size is too small.

D. Can't be calculated because the sample size is too small.

# VIII. Hypothesis Testing

## Fundamentals of Hypothesis Testing

1. Suppose you were hired to conduct a study to find out which of two brands of soda college students think taste better. In your study, students are given a blind taste test. They rate one brand and then rated the other, in random order. The ratings are given on a scale of 1 (awful) to 5 (delicious). Which type of test would be the best to compare these ratings?

A. One-Sample t

B. Chi-Square

C. Paired Difference t

D. Two-Sample t

2. USA Today's AD Track examined the effectiveness of the new ads involving the Pets.com Sock Puppet (which is now extinct). In particular, they conducted a nationwide poll of 428 adults who had seen the Pets.com ads and asked for their opinions. They found that 36% of the respondents said they liked the ads. Suppose you increased the sample size for this poll to 1000, but you had the same sample percentage who like the ads (36%). How would this change the p-value of the hypothesis test you want to conduct?

A. No way to tell

B. The new p-value would be the same as before

C. The new p-value would be smaller than before

D. The new p-value would be larger than before

3. A marketing director for a radio station collects a random sample of three hundred 18 to 25 year-olds and two hundred and fifty 25 to 40 year-olds. She records the percent of each group that had purchased music online in the last 30 days. She performs a hypothesis test, and the p-value of her test turns out to be 0.15. From this she should conclude:

A. that about 15% more people purchased on-line music in the younger group than in the older group.

B. there is insufficient evidence to conclude that there is a difference in the proportion of on-line music purchases in the younger and older group.

C. the proportion of on-line music purchasers is the same in the under-25 year-old group as in the older group.

D. the probability of getting the same results again is 0.15.

4. If we want to estimate the mean difference in scores on a pre-test and post-test for a sample of students, how should we proceed?

A. We should construct a confidence interval or conduct a hypothesis test

B. We should collect one sample, two samples, or conduct a paired data procedure

C. We should calculate a z or a t statistic

5. The paint used to make lines on roads must reflect enough light to be clearly visible at night. Let mu denote the true average reflectometer reading for a new type of paint under consideration. A test of the null hypothesis that mu = 20 versus the alternative hypothesis that mu > 20 will be based on a random sample of size n from a normal population distribution. In which of the following scenarios is there significant evidence that mu is larger than 20?

(i) n=15, t=3.2, alpha=0.05

(ii) n=9, t=1.8, alpha=0.01

(iii) n=24, t=-0.2, alpha=0.01

A. (ii) and (iii)

B. (i)

C. (iii)

D. (ii)

## Testing a Claim About a Mean: Large Samples

1. Hong is a pharmacist studying the effect of an anti-depressant drug. She organizes a simple random sample of 100 patients, and then collect their anxiety test scores before and after administering the anti-depressant drug. Hong wants to estimate the mean difference between the pre-drug and post-drug test scores. How should she proceed?

A. She should compute a confidence interval or conduct a hypothesis test

B. She should calculate the z or the t statistics

C. She should compute the correlation between the two samples

D. Not enough information to tell

# X. Correlation and regression

## Correlation

1. A positive correlation between two variables X and Y means that if X increases, this will cause the value of Y to increase.

A. This is always true.

B. This is sometimes true.

C. This is never true.

2. The correlation between high school algebra and geometry scores was found to be + 0.8. Which of the following statements is not true?

A. Most of the students who have above average scores in algebra also have above average scores in geometry.

B. Most people who have above average scores in algebra will have below average scores in geometry

C. If we increase a student's score in algebra (ie. with extra tutoring in algebra), then the student's geometry scores will always increase accordingly.

D. Most students who have below average scores in algebra also have below average scores in geometry.