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AP® Statistics

4th Edition

Freedman, Robert Pisani, Roger Purves

ISBN: 9780393929720

Textbook solutions

Chapter 2 :, chapter 3 :, chapter 4 :, chapter 5 :, chapter 6 :, chapter 7 :, chapter 8 :, chapter 9 :, chapter 10 :, chapter 11 :, chapter 12 :, chapter 13 :, chapter 14 :, chapter 15 :, chapter 16 :, chapter 17 :, chapter 18 :, chapter 19 :, chapter 20 :, chapter 21 :, chapter 22 :, chapter 23 :, chapter 24 :, chapter 25 :, chapter 26 :, chapter 27 :, chapter 28 :, chapter 29 :.

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mean = 4 hours, standard deviation = 1.2 hours, sample size = 16

a. Check student's solution. b. 3.5, 4.25, 0.2441

The fact that the two distributions are different accounts for the different probabilities.

The sample size, n , gets larger.

• U (24, 26), 25, 0.5774
• N (25, 0.0577)
• N (2500, 5.7735)

N ( 10,  10 8 ) ( 10,  10 8 )

• Χ = amount of change students carry
• Χ ~ Exp (1/0.88) or approximately Χ ~ Exp (1.1364)
• X ¯ X ¯ = average amount of change carried by a sample of 25 students.
• X ¯ X ¯ ~ N (0.88, 0.176)
• The probability in part (e) represents the probability of an individual value. In part (f), the probability describes the mean of a sample of 25. Part (f) relies on the central limit theorem, so the distributions are different. Part (e) is exponential and part (f) is normal.
• length of time for an individual to complete IRS form 1040, in hours
• mean length of time for a sample of 36 taxpayers to complete IRS form 1040, in hours
• N ( 10 .53,  1 3 ) ( 10 .53,  1 3 )
• Yes, I would be surprised, because the probability is almost 0.
• No, I would not be totally surprised because the probability is 0.2312.
• the length of a song, in minutes, in the collection
• the average length, in minutes, of the songs from a sample of five albums from the collection
• N (2.75, 0.0220)
• 2.74 minutes
• 0.03 minutes
• True. The mean of a sampling distribution of the means is approximately the mean of the data distribution.
• True. According to the central limit theorem, the larger the sample, the closer the sampling distribution of the means becomes normal.
• The standard deviation of the sampling distribution of the means will decrease, making it approximately the same as the standard deviation of X as the sample size increases.
• X = the yearly income of someone in a Third World country
• the average salary from samples of 1,000 residents of a Third World country
• X ¯ X ¯ ∼ N ( 2,000,  8,000 1,000 ) ( 2,000,  8,000 1,000 )
• Very wide differences in data values can have averages smaller than standard deviations.
• The distribution of the sample mean will have higher probabilities closer to the population mean. P (2,000 < X ¯ X ¯ < 2,100) = 0.1537 P (2,100 < X ¯ X ¯ < 2,200) = 0.1317
• the total length of time for nine criminal trials
• N (189, 21)
• 162.09; 90 percent of the total nine trials of this type will last 162 days or more.
• X = the salary of one elementary school teacher in the district
• X ~ N (44000, 6500)
• ΣX ~ sum of the salaries of 10 elementary school teachers in the sample
• ΣX ~ N (44,000, 20,554.80)
• Sampling 70 teachers instead of 10 would cause the distribution to be more spread out. It would be a more symmetrical normal curve.
• If every teacher received a $3,000 raise, the distribution of X would shift to the right by $3,000. In other words, it would have a mean of $47,000.
• X = the closing stock prices for U.S. semiconductor manufacturers
• i. $20.71, ii. $17.31, iii. 35
• exponential distribution, Χ ~ Exp ( 1 20.71 ) ( 1 20.71 )
• Answers will vary.
• i. $20.71, ii. $11.14
• N ( 20 .71,  17.31 5 ) ( 20 .71,  17.31 5 )
• Check student’s solution.
• X ¯ X ¯ ~ N ( 60,  9 25 ) ( 60,  9 25 )
• N (1500, 45)
• We have μ = 17, σ = 0.8, x ¯ x ¯ = 16.7, and n = 30. To calculate the probability, we use normalcdf (lower, upper, μ , σ n σ n ) = normalcdf ( E – 99,16 .7,17, 0. 8 30 ) ( E – 99,16 .7,17, 0. 8 30 ) = 0.0200.
• If the process is working properly, then the probability that a sample of 30 batteries would have at most 16.7 life span hours is only 2%. Therefore, the class was justified to question the claim.
• For the sample, we have n = 100, x ¯ x ¯ = 0.862, and s = 0.05.
• Σ x ¯ Σ x ¯ = 85.65, Σs = 5.18
• normalcdf (396.9, E 99,(465)(0.8565),(0.05)( 465 465 )) ≈ 1
• Because the probability of a sample of size of 465 having at least a mean sum of 396.9 is appproximately 1, we can conclude that the company is correctly labeling their candy packages.

Use normalcdf ( E – 99,1 .1,1, 1 70 ) ( E – 99,1 .1,1, 1 70 ) = 0.7986. This means that there is an 80 percent chance that the service time will be less than 1.1 hours. It may be wise to schedule more time because there is an associated 20 percent chance that the maintenance time will be greater than 1.1 hours.

Because we have normalcdf ( 5. 111,5 .291,5 .201, 0. 065 280 ) ( 5. 111,5 .291,5 .201, 0. 065 280 ) ≈ 1, we can conclude that practically all the coins are within the limits; therefore, there should be no rejected coins out of a well-selected sample size of 280.

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1.E: Introduction to Statistics (Exercises)

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These are homework exercises to accompany the Textmap created for "Introductory Statistics" by Shafer and Zhang.

1.1: Basic Definitions and Concepts

Explain what is meant by the term population .

Explain what is meant by the term sample .

Explain how a sample differs from a population.

Explain what is meant by the term sample data .

Explain what a parameter is.

Explain what a statistic is.

Give an example of a population and two different characteristics that may be of interest.

Describe the difference between descriptive statistics and inferential statistics . Illustrate with an example.

Identify each of the following data sets as either a population or a sample:

• The grade point averages (GPAs) of all students at a college.
• The GPAs of a randomly selected group of students on a college campus.
• The ages of the nine Supreme Court Justices of the United States on \(\text{January}\; 1,\; 1842\).
• The gender of every second customer who enters a movie theater.
• The lengths of Atlantic croakers caught on a fishing trip to the beach.

Identify the following measures as either quantitative or qualitative:

• The \(30\) high-temperature readings of the last \(30\) days.
• The scores of \(40\) students on an English test.
• The blood types of \(120\) teachers in a middle school.
• The last four digits of social security numbers of all students in a class.
• The numbers on the jerseys of \(53\) football players on a team.
• The genders of the first \(40\) newborns in a hospital one year.
• The natural hair color of \(20\) randomly selected fashion models.
• The ages of \(20\) randomly selected fashion models.
• The fuel economy in miles per gallon of \(20\) new cars purchased last month.
• The political affiliation of \(500\) randomly selected voters.

A researcher wishes to estimate the average amount spent per person by visitors to a theme park. He takes a random sample of forty visitors and obtains an average of \(\$28\) per person.

• What is the population of interest?
• What is the parameter of interest?
• Based on this sample, do we know the average amount spent per person by visitors to the park? Explain fully.

A researcher wishes to estimate the average weight of newborns in South America in the last five years. He takes a random sample of \(235\) newborns and obtains an average of \(3.27\) kilograms.

• Based on this sample, do we know the average weight of newborns in South America? Explain fully.

A researcher wishes to estimate the proportion of all adults who own a cell phone. He takes a random sample of \(1,572\) adults; \(1,298\) of them own a cell phone, hence \(1298/1572 \approx 0.83\) or about \(83\%\) own a cell phone.

• What is the statistic involved?
• Based on this sample, do we know the proportion of all adults who own a cell phone? Explain fully.

A sociologist wishes to estimate the proportion of all adults in a certain region who have never married. In a random sample of \(1,320\) adults, \(145\) have never married, hence \(145/1320 \approx 0.11\) or about \(11\%\) have never married.

• Based on this sample, do we know the proportion of all adults who have never married? Explain fully.
• What must be true of a sample if it is to give a reliable estimate of the value of a particular population parameter?
• What must be true of a sample if it is to give certain knowledge of the value of a particular population parameter?
• A population is the total collection of objects that are of interest in a statistical study.
• A sample, being a subset, is typically smaller than the population. In a statistical study, all elements of a sample are available for observation, which is not typically the case for a population.
• A parameter is a value describing a characteristic of a population. In a statistical study the value of a parameter is typically unknown.
• All currently registered students at a particular college form a population. Two population characteristics of interest could be the average GPA and the proportion of students over \(23\) years.
• Population.
• Qualitative.
• Quantitative.
• All newborn babies in South America in the last five years.
• The average birth weight of all newborn babies in South America in the last five years.
• No, not exactly, but we know the approximate value of the average.
• All adults in the region.
• The proportion of the adults in the region who have never married.
• The proportion computed from the sample, \(0.1\).
• No, not exactly, but we know the approximate value of the proportion.

1.2: Overview

1.3: presentation of data.

1. List all the measurements for the data set represented by the following data frequency table.

\[\begin{array}{c|ccccc}x & 31 & 32 & 33 & 34 & 35 \\ \hline f & 1 & 5 & 6 & 4 & 2\end{array}\]

2. List all the measurements for the data set represented by the following data frequency table

\[\begin{array}{c|ccccccc}x & 97 & 98 & 99 & 100 & 101 & 102 & 103 & 105 \\ \hline f & 7 & 5 & 3 & 4 & 2 & 2 & 1 & 1\end{array}\]

3. Construct the data frequency table for the following data set.

\[\begin{array}22 & 25 & 22 & 27 & 24 & 23 \\ 26 & 24 & 22 & 24 & 26 &\end{array}\]

4. Construct the data frequency table for the following data set.

\[ \{1,\, 5,\, 2,\, 3,\, 5,\, 1,\, 4,\, 4,\, 4,\, 3,\, 2,\, 5,\, 1,\, 3,\, 2,\, 1,\, 1,\, 1,\, 2\} \]

• \(\{31,\, 32,\, 32,\, 32,\, 32,\, 32,\, 33,\, 33,\, 33,\, 33,\, 33,\, 33,\, 34,\, 34,\, 34,\, 34,\, 35,\, 35\}\)
• \(\begin{array}{c|ccccc}x & 22 & 23 & 24 & 25 & 26 & 27 \\ \hline f & 3 & 1 & 3 & 1 & 2 & 1\end{array}\)

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Statistics and Probability Worksheets

Welcome to the statistics and probability page at Math-Drills.com where there is a 100% chance of learning something! This page includes Statistics worksheets including collecting and organizing data, measures of central tendency (mean, median, mode and range) and probability.

Students spend their lives collecting, organizing, and analyzing data, so why not teach them a few skills to help them on their way. Data management is probably best done on authentic tasks that will engage students in their own learning. They can collect their own data on topics that interest them. For example, have you ever wondered if everyone shares the same taste in music as you? Perhaps a survey, a couple of graphs and a few analysis sentences will give you an idea.

Statistics has applications in many different fields of study. Budding scientists, stock market brokers, marketing geniuses, and many other pursuits will involve managing data on a daily basis. Teaching students critical thinking skills related to analyzing data they are presented will enable them to make crucial and informed decisions throughout their lives.

Probability is a topic in math that crosses over to several other skills such as decimals, percents, multiplication, division, fractions, etc. Probability worksheets will help students to practice all of these skills with a chance of success!

Most Popular Statistics and Probability Worksheets this Week

Mean, Median, Mode and Range Worksheets

Calculating the mean, median, mode and range are staples of the upper elementary math curriculum. Here you will find worksheets for practicing the calculation of mean, median, mode and range. In case you're not familiar with these concepts, here is how to calculate each one. To calculate the mean, add all of the numbers in the set together and divide that sum by the number of numbers in the set. To calculate the median, first arrange the numbers in order, then locate the middle number. In sets where there are an even number of numbers, calculate the mean of the two middle numbers. To calculate the mode, look for numbers that repeat. If there is only one of each number, the set has no mode. If there are doubles of two different numbers and there are more numbers in the set, the set has two modes. If there are triples of three different numbers and there are more numbers in the set, the set has three modes, and so on. The range is calculated by subtracting the least number from the greatest number.

Note that all of the measures of central tendency are included on each page, but you don't need to assign them all if you aren't working on them all. If you're only working on mean, only assign students to calculate the mean.

In order to determine the median, it is necessary to have your numbers sorted. It is also helpful in determining the mode and range. To expedite the process, these first worksheets include the lists of numbers already sorted.

• Calculating Mean, Median, Mode and Range from Sorted Lists Sets of 5 Numbers from 1 to 10 Sets of 5 Numbers from 10 to 99 Sets of 5 Numbers from 100 to 999 Sets of 10 Numbers from 1 to 10 Sets of 10 Numbers from 10 to 99 Sets of 10 Numbers from 100 to 999 Sets of 20 Numbers from 10 to 99 Sets of 15 Numbers from 100 to 999

Normally, data does not come in a sorted list, so these worksheets are a little more realistic. To find some of the statistics, it will be easier for students to put the numbers in order first.

• Calculating Mean, Median, Mode and Range from Unsorted Lists Sets of 5 Numbers from 1 to 10 Sets of 5 Numbers from 10 to 99 Sets of 5 Numbers from 100 to 999 Sets of 10 Numbers from 1 to 10 Sets of 10 Numbers from 10 to 99 Sets of 10 Numbers from 100 to 999 Sets of 20 Numbers from 10 to 99 Sets of 15 Numbers from 100 to 999

Collecting and Organizing Data

Teaching students how to collect and organize data enables them to develop skills that will enable them to study topics in statistics with more confidence and deeper understanding.

• Constructing Line Plots from Small Data Sets Construct Line Plots with Smaller Numbers and Lines with Ticks Provided (Small Data Set) Construct Line Plots with Smaller Numbers and Lines Only Provided (Small Data Set) Construct Line Plots with Smaller Numbers (Small Data Set) Construct Line Plots with Larger Numbers and Lines with Ticks Provided (Small Data Set) Construct Line Plots with Larger Numbers and Lines Only Provided (Small Data Set) Construct Line Plots with Larger Numbers (Small Data Set)
• Constructing Line Plots from Larger Data Sets Construct Line Plots with Smaller Numbers and Lines with Ticks Provided Construct Line Plots with Smaller Numbers and Lines Only Provided Construct Line Plots with Smaller Numbers Construct Line Plots with Larger Numbers and Lines with Ticks Provided Construct Line Plots with Larger Numbers and Lines Only Provided Construct Line Plots with Larger Numbers

Interpreting and Analyzing Data

Answering questions about graphs and other data helps students build critical thinking skills. Standard questions include determining the minimum, maximum, range, count, median, mode, and mean.

• Answering Questions About Stem-and-Leaf Plots Stem-and-Leaf Plots with about 25 data points Stem-and-Leaf Plots with about 50 data points Stem-and-Leaf Plots with about 100 data points
• Answering Questions About Line Plots Line Plots with Smaller Data Sets and Smaller Numbers Line Plots with Smaller Data Sets and Larger Numbers Line Plots with Larger Data Sets and Smaller Numbers Line Plots with Larger Data Sets and Larger Numbers
• Answering Questions About Broken-Line Graphs Answer Questions About Broken-Line Graphs
• Answering Questions About Circle Graphs Circle Graph Questions (Color Version) Circle Graph Questions (Black and White Version) Circle Graphs No Questions (Color Version) Circle Graphs No Questions (Black and White Version)
• Answering Questions About Pictographs Answer Questions About Pictographs

Probability Worksheets

• Calculating Probabilities with Dice Sum of Two Dice Probabilities Sum of Two Dice Probabilities (with table)

Spinners can be used for probability experiments or for theoretical probability. Students should intuitively know that a number that is more common on a spinner will come up more often. Spinning 100 or more times and tallying the results should get them close to the theoretical probability. The more sections there are, the more spins will be needed.

• Calculating Probabilities with Number Spinners Number Spinner Probability (4 Sections) Number Spinner Probability (5 Sections) Number Spinner Probability (6 Sections) Number Spinner Probability (7 Sections) Number Spinner Probability (8 Sections) Number Spinner Probability (9 Sections) Number Spinner Probability (10 Sections) Number Spinner Probability (11 Sections) Number Spinner Probability (12 Sections)

Non-numerical spinners can be used for experimental or theoretical probability. There are basic questions on every version with a couple extra questions on the A and B versions. Teachers and students can make up other questions to ask and conduct experiments or calculate the theoretical probability. Print copies for everyone or display on an interactive white board.

• Probability with Single-Event Spinners Animal Spinner Probability ( 4 Sections) Animal Spinner Probability ( 5 Sections) Animal Spinner Probability ( 10 Sections) Letter Spinner Probability ( 4 Sections) Letter Spinner Probability ( 5 Sections) Letter Spinner Probability ( 10 Sections) Color Spinner Probability ( 4 Sections) Color Spinner Probability ( 5 Sections) Color Spinner Probability ( 10 Sections)
• Probability with Multi-Event Spinners Animal/Letter Combined Spinner Probability ( 4 Sections) Animal/Letter Combined Spinner Probability ( 5 Sections) Animal/Letter Combined Spinner Probability ( 10 Sections) Animal/Letter/Color Combined Spinner Probability ( 4 Sections) Animal/Letter/Color Combined Spinner Probability ( 5 Sections) Animal/Letter/Color Combined Spinner Probability ( 10 Sections)

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Teach yourself statistics

Statistics Problems

One of the best ways to learn statistics is to solve practice problems. These problems test your understanding of statistics terminology and your ability to solve common statistics problems. Each problem includes a step-by-step explanation of the solution.

• Use the dropdown boxes to describe the type of problem you want to work on.
• click the Submit button to see problems and solutions.

Main topic:

Problem description:

In one state, 52% of the voters are Republicans, and 48% are Democrats. In a second state, 47% of the voters are Republicans, and 53% are Democrats. Suppose a simple random sample of 100 voters are surveyed from each state.

What is the probability that the survey will show a greater percentage of Republican voters in the second state than in the first state?

The correct answer is C. For this analysis, let P 1 = the proportion of Republican voters in the first state, P 2 = the proportion of Republican voters in the second state, p 1 = the proportion of Republican voters in the sample from the first state, and p 2 = the proportion of Republican voters in the sample from the second state. The number of voters sampled from the first state (n 1 ) = 100, and the number of voters sampled from the second state (n 2 ) = 100.

The solution involves four steps.

• Make sure the sample size is big enough to model differences with a normal population. Because n 1 P 1 = 100 * 0.52 = 52, n 1 (1 - P 1 ) = 100 * 0.48 = 48, n 2 P 2 = 100 * 0.47 = 47, and n 2 (1 - P 2 ) = 100 * 0.53 = 53 are each greater than 10, the sample size is large enough.
• Find the mean of the difference in sample proportions: E(p 1 - p 2 ) = P 1 - P 2 = 0.52 - 0.47 = 0.05.

σ d = sqrt{ [ P1( 1 - P 1 ) / n 1 ] + [ P 2 (1 - P 2 ) / n 2 ] }

σ d = sqrt{ [ (0.52)(0.48) / 100 ] + [ (0.47)(0.53) / 100 ] }

σ d = sqrt (0.002496 + 0.002491) = sqrt(0.004987) = 0.0706

z p 1 - p 2 = (x - μ p 1 - p 2 ) / σ d = (0 - 0.05)/0.0706 = -0.7082

Using Stat Trek's Normal Distribution Calculator , we find that the probability of a z-score being -0.7082 or less is 0.24.

Therefore, the probability that the survey will show a greater percentage of Republican voters in the second state than in the first state is 0.24.

See also: Difference Between Proportions

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1.4: Introduction to Statistics (Exercises)

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These are homework exercises to accompany the Textmap created for "Introductory Statistics" by Shafer and Zhang. Complementary General Chemistry question banks can be found for other Textmaps and can be accessed here . In addition to these publicly available questions, access to private problems bank for use in exams and homework is available to faculty only on an individual basis; please contact Delmar Larsen for an account with access permission.

1.1: Basic Definitions and Concepts

• Explain what is meant by the term population .
• Explain what is meant by the term sample .
• Explain how a sample differs from a population.
• Explain what is meant by the term sample data .
• Explain what a parameter is.
• Explain what a statistic is.
• Give an example of a population and two different characteristics that may be of interest.
• Describe the difference between descriptive statistics and inferential statistics . Illustrate with an example.
• The grade point averages (GPAs) of all students at a college.
• The GPAs of a randomly selected group of students on a college campus.
• The ages of the nine Supreme Court Justices of the United States on \(\text {January}\: 1, 1842\).
• The gender of every second customer who enters a movie theater.
• The lengths of Atlantic croakers caught on a fishing trip to the beach.
• The \(30\) high-temperature readings of the last \(30\) days.
• The scores of \(40\) students on an English test.
• The blood types of \(120\) teachers in a middle school.
• The last four digits of social security numbers of all students in a class.
• The numbers on the jerseys of \(53\) football players on a team.
• The genders of the first \(40\) newborns in a hospital one year.
• The natural hair color of \(20\) randomly selected fashion models.
• The ages of \(20\) randomly selected fashion models.
• The fuel economy in miles per gallon of \(20\) new cars purchased last month.
• The political affiliation of \(500\) randomly selected voters.
• What is the population of interest?
• What is the parameter of interest?
• Based on this sample, do we know the average amount spent per person by visitors to the park? Explain fully.
• Based on this sample, do we know the average weight of newborns in South America? Explain fully.
• What is the statistic involved?
• Based on this sample, do we know the proportion of all adults who own a cell phone? Explain fully.
• Based on this sample, do we know the proportion of all adults who have never married? Explain fully.
• What must be true of a sample if it is to give a reliable estimate of the value of a particular population parameter?
• What must be true of a sample if it is to give certain knowledge of the value of a particular population parameter?
• A population is the total collection of objects that are of interest in a statistical study.
• A sample, being a subset, is typically smaller than the population. In a statistical study, all elements of a sample are available for observation, which is not typically the case for a population.
• A parameter is a value describing a characteristic of a population. In a statistical study the value of a parameter is typically unknown.
• All currently registered students at a particular college form a population. Two population characteristics of interest could be the average GPA and the proportion of students over \(23\) years.
• Population.
• Qualitative.
• Quantitative.
• All newborn babies in South America in the last five years.
• The average birth weight of all newborn babies in South America in the last five years.
• No, not exactly, but we know the approximate value of the average.
• All adults in the region.
• The proportion of the adults in the region who have never married.
• The proportion computed from the sample, \(0.1\).
• No, not exactly, but we know the approximate value of the proportion.

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