statistics 5 2 homework

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AP®︎/College Statistics

Unit 1: exploring categorical data, unit 2: exploring one-variable quantitative data: displaying and describing, unit 3: exploring one-variable quantitative data: summary statistics, unit 4: exploring one-variable quantitative data: percentiles, z-scores, and the normal distribution, unit 5: exploring two-variable quantitative data, unit 6: collecting data, unit 7: probability, unit 8: random variables and probability distributions, unit 9: sampling distributions, unit 10: inference for categorical data: proportions, unit 11: inference for quantitative data: means, unit 12: inference for categorical data: chi-square, unit 13: inference for quantitative data: slopes, unit 14: prepare for the 2022 ap®︎ statistics exam.

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5.1: Introduction

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• Page ID 20036

CHAPTER OBJECTIVES

By the end of this chapter, the student should be able to:

• Recognize and understand discrete probability distribution functions, in general.
• Calculate and interpret expected values.
• Recognize and understand continuous probability density functions in general.
• Recognize the normal probability distribution.
• A student takes a ten-question, true-false quiz. Because the student had such a busy schedule, he or she could not study and guesses randomly at each answer. What is the probability of the student passing the test with at least a 70%?
• Small companies might be interested in the number of long-distance phone calls their employees make during the peak time of the day. Suppose the average is 20 calls. What is the probability that the employees make more than 20 long-distance phone calls during the peak time?

These two examples illustrate two different types of probability problems involving discrete random variables. Recall that discrete data are data that you can count. A random variable describes the outcomes of a statistical experiment in words. The values of a random variable can vary with each repetition of an experiment.

Random Variable Notation

Upper case letters such as \(X\) or \(Y\) denote a random variable. Lower case letters like \(x\) or \(y\) denote the value of a random variable. If \(X\) is a random variable, then \(X\) is written in words, and x is given as a number.

For example, let \(X =\) the number of heads you get when you toss three fair coins. The sample space for the toss of three fair coins is TTT ; THH ; HTH ; HHT ; HTT ; THT ; TTH ; HHH . Then, \(x =\) 0, 1, 2, 3. \(X\) is in words and x is a number. Notice that for this example, the \(x\) values are countable outcomes. Because you can count the possible values that \(X\) can take on and the outcomes are random (the x values 0, 1, 2, 3), \(X\) is a discrete random variable.

Probability Distributions

Continuous random variables have many applications. Baseball batting averages, IQ scores, the length of time a long distance telephone call lasts, the amount of money a person carries, the length of time a computer chip lasts, and SAT scores are just a few. The field of reliability depends on a variety of continuous random variables.

The values of discrete and continuous random variables can be ambiguous. For example, if \(X\) is equal to the number of miles (to the nearest mile) you drive to work, then \(X\) is a discrete random variable. You count the miles. If \(X\) is the distance you drive to work, then you measure values of \(X\) and \(X\) is a continuous random variable. For a second example, if \(X\) is equal to the number of books in a backpack, then \(X\) is a discrete random variable. If \(X\) is the weight of a book, then \(X\) is a continuous random variable because weights are measured. How the random variable is defined is very important.

Properties of Continuous Probability Distributions

The graph of a continuous probability distribution is a curve. Probability is represented by area under the curve. The curve is called the probability density function (abbreviated as pdf). We use the symbol \(f(x)\) to represent the curve. \(f(x)\) is the function that corresponds to the graph; we use the density function \(f(x)\) to draw the graph of the probability distribution. Area under the curve is given by a different function called the cumulative distribution function(abbreviated as cdf). The cumulative distribution function is used to evaluate probability as area.

• The outcomes are measured, not counted.
• The entire area under the curve and above the x-axis is equal to one.
• Probability is found for intervals of \(x\) values rather than for individual \(x\) values.
• \(P(c < x < d)\) is the probability that the random variable \(X\) is in the interval between the values \(c\) and \(d\). \(P(c < x < d)\) is the area under the curve, above the x -axis, to the right of \(c\) and the left of \(d\).
• \(P(x = c) = 0\) The probability that \(x\) takes on any single individual value is zero. The area below the curve, above the x -axis, and between \(x = c\) and \(x = c\) has no width, and therefore no area (area = 0). Since the probability is equal to the area, the probability is also zero.
• \(P(c < x < d)\) is the same as \(P(c \leq x \leq d)\) because probability is equal to area.

We will find the area that represents probability by using geometry, formulas, technology, or probability tables. In general, calculus is needed to find the area under the curve for many probability density functions. When we use formulas to find the area in this textbook, the formulas were found by using the techniques of integral calculus. However, because most students taking this course have not studied calculus, we will not be using calculus in this textbook. There are many continuous probability distributions. When using a continuous probability distribution to model probability, the distribution used is selected to model and fit the particular situation in the best way.

In this chapter and the next, we will study the uniform distribution, the exponential distribution, and the normal distribution. The following graphs illustrate these distributions.

Collaborative Exercise

Toss a coin ten times and record the number of heads. After all members of the class have completed the experiment (tossed a coin ten times and counted the number of heads), fill in Table . Let \(X =\) the number of heads in ten tosses of the coin.

• Which value(s) of \(x\) occurred most frequently?
• If you tossed the coin 1,000 times, what values could \(x\) take on? Which value(s) of \(x\) do you think would occur most frequently?
• What does the relative frequency column sum to?

Contributors and Attributions

Barbara Illowsky and Susan Dean (De Anza College) with many other contributing authors. Content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at http://cnx.org/contents/[email protected] .

11 Surprising Homework Statistics, Facts & Data

The age-old question of whether homework is good or bad for students is unanswerable because there are so many “ it depends ” factors.

For example, it depends on the age of the child, the type of homework being assigned, and even the child’s needs.

There are also many conflicting reports on whether homework is good or bad. This is a topic that largely relies on data interpretation for the researcher to come to their conclusions.

To cut through some of the fog, below I’ve outlined some great homework statistics that can help us understand the effects of homework on children.

Homework Statistics List

1. 45% of parents think homework is too easy for their children.

A study by the Center for American Progress found that parents are almost twice as likely to believe their children’s homework is too easy than to disagree with that statement.

Here are the figures for math homework:

• 46% of parents think their child’s math homework is too easy.
• 25% of parents think their child’s math homework is not too easy.
• 29% of parents offered no opinion.

Here are the figures for language arts homework:

• 44% of parents think their child’s language arts homework is too easy.
• 28% of parents think their child’s language arts homework is not too easy.
• 28% of parents offered no opinion.

These findings are based on online surveys of 372 parents of school-aged children conducted in 2018.

2. 93% of Fourth Grade Children Worldwide are Assigned Homework

The prestigious worldwide math assessment Trends in International Maths and Science Study (TIMSS) took a survey of worldwide homework trends in 2007. Their study concluded that 93% of fourth-grade children are regularly assigned homework, while just 7% never or rarely have homework assigned.

3. 17% of Teens Regularly Miss Homework due to Lack of High-Speed Internet Access

A 2018 Pew Research poll of 743 US teens found that 17%, or almost 2 in every 5 students, regularly struggled to complete homework because they didn’t have reliable access to the internet.

This figure rose to 25% of Black American teens and 24% of teens whose families have an income of less than $30,000 per year.

4. Parents Spend 6.7 Hours Per Week on their Children’s Homework

A 2018 study of 27,500 parents around the world found that the average amount of time parents spend on homework with their child is 6.7 hours per week. Furthermore, 25% of parents spend more than 7 hours per week on their child’s homework.

American parents spend slightly below average at 6.2 hours per week, while Indian parents spend 12 hours per week and Japanese parents spend 2.6 hours per week.

5. Students in High-Performing High Schools Spend on Average 3.1 Hours per night Doing Homework

A study by Galloway, Conner & Pope (2013) conducted a sample of 4,317 students from 10 high-performing high schools in upper-middle-class California. 

Across these high-performing schools, students self-reported that they did 3.1 hours per night of homework.

Graduates from those schools also ended up going on to college 93% of the time.

6. One to Two Hours is the Optimal Duration for Homework

A 2012 peer-reviewed study in the High School Journal found that students who conducted between one and two hours achieved higher results in tests than any other group.

However, the authors were quick to highlight that this “t is an oversimplification of a much more complex problem.” I’m inclined to agree. The greater variable is likely the quality of the homework than time spent on it.

Nevertheless, one result was unequivocal: that some homework is better than none at all : “students who complete any amount of homework earn higher test scores than their peers who do not complete homework.”

7. 74% of Teens cite Homework as a Source of Stress

A study by the Better Sleep Council found that homework is a source of stress for 74% of students. Only school grades, at 75%, rated higher in the study.

That figure rises for girls, with 80% of girls citing homework as a source of stress.

Similarly, the study by Galloway, Conner & Pope (2013) found that 56% of students cite homework as a “primary stressor” in their lives.

8. US Teens Spend more than 15 Hours per Week on Homework

The same study by the Better Sleep Council also found that US teens spend over 2 hours per school night on homework, and overall this added up to over 15 hours per week.

Surprisingly, 4% of US teens say they do more than 6 hours of homework per night. That’s almost as much homework as there are hours in the school day.

The only activity that teens self-reported as doing more than homework was engaging in electronics, which included using phones, playing video games, and watching TV.

9. The 10-Minute Rule

The National Education Association (USA) endorses the concept of doing 10 minutes of homework per night per grade.

For example, if you are in 3rd grade, you should do 30 minutes of homework per night. If you are in 4th grade, you should do 40 minutes of homework per night.

However, this ‘rule’ appears not to be based in sound research. Nevertheless, it is true that homework benefits (no matter the quality of the homework) will likely wane after 2 hours (120 minutes) per night, which would be the NEA guidelines’ peak in grade 12.

10. 21.9% of Parents are Too Busy for their Children’s Homework

An online poll of nearly 300 parents found that 21.9% are too busy to review their children’s homework. On top of this, 31.6% of parents do not look at their children’s homework because their children do not want their help. For these parents, their children’s unwillingness to accept their support is a key source of frustration.

11. 46.5% of Parents find Homework too Hard

The same online poll of parents of children from grades 1 to 12 also found that many parents struggle to help their children with homework because parents find it confusing themselves. Unfortunately, the study did not ask the age of the students so more data is required here to get a full picture of the issue.

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Interpreting the Data

Unfortunately, homework is one of those topics that can be interpreted by different people pursuing differing agendas. All studies of homework have a wide range of variables, such as:

• What age were the children in the study?
• What was the homework they were assigned?
• What tools were available to them?
• What were the cultural attitudes to homework and how did they impact the study?
• Is the study replicable?

The more questions we ask about the data, the more we realize that it’s hard to come to firm conclusions about the pros and cons of homework .

Furthermore, questions about the opportunity cost of homework remain. Even if homework is good for children’s test scores, is it worthwhile if the children consequently do less exercise or experience more stress?

Thus, this ends up becoming a largely qualitative exercise. If parents and teachers zoom in on an individual child’s needs, they’ll be able to more effectively understand how much homework a child needs as well as the type of homework they should be assigned.

Related: Funny Homework Excuses

The debate over whether homework should be banned will not be resolved with these homework statistics. But, these facts and figures can help you to pursue a position in a school debate on the topic – and with that, I hope your debate goes well and you develop some great debating skills!

Chris Drew (PhD)

Dr. Chris Drew is the founder of the Helpful Professor. He holds a PhD in education and has published over 20 articles in scholarly journals. He is the former editor of the Journal of Learning Development in Higher Education. [Image Descriptor: Photo of Chris]

• Chris Drew (PhD) https://helpfulprofessor.com/author/chris-drew-phd/ 5 Top Tips for Succeeding at University
• Chris Drew (PhD) https://helpfulprofessor.com/author/chris-drew-phd/ 50 Durable Goods Examples
• Chris Drew (PhD) https://helpfulprofessor.com/author/chris-drew-phd/ 100 Consumer Goods Examples
• Chris Drew (PhD) https://helpfulprofessor.com/author/chris-drew-phd/ 30 Globalization Pros and Cons

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Helping math teachers bring statistics to life

Random Variables

Chapter 5 Lesson Plans

Chapter 1 Chapter 2

Eoy project.

Day 1: Lesson 5.1 - Introduction to Random Variables Day 2: Lesson 5.2 - Analyzing Discrete Random Variables

Day 3: Mathalicious - Basketball IQ Day 4: Mathalicious - Three Shots Day 5: Lesson 5.3 - Binomial Random Variables Day 6: Lesson 5.4 - Analyzing Binomial Random Variables Day 7: Quiz 5.1 to 5.4 Day 8: Lesson 5.5 - Continuous Random Variables Day 9: Lesson 5.6 - The Standard Normal Distribution Day 10: Lesson 5.7 - Normal Distributions Calculations Day 11: Normal Distributions Practice Day 12: Quiz 5.5 to 5.7 Day 13: Chapter 5 Review Day 14: Normal Distribution Foldable Review Day 15: Chapter 5 Test

Chapter 2: Descriptive Statistics

Chapter 2 Homework

Homework from 2.1.

Student grades on a chemistry exam were: 77, 78, 76, 81, 86, 51, 79, 82, 84, 99

• Construct a stem-and-leaf plot of the data.
• Are there any potential outliers? If so, which scores are they? Why do you consider them outliers?

[link] contains the 2010 obesity rates in U.S. states and Washington, DC.

• Use a random number generator to randomly pick eight states. Construct a bar graph of the obesity rates of those eight states.
• Construct a bar graph for all the states beginning with the letter “A.”
• Construct a bar graph for all the states beginning with the letter “M.”
• Number the entries in the table 1–51 (Includes Washington, DC; Numbered vertically)
• Arrow over to PRB
• Press 5:randInt(
• Enter 51,1,8)

Eight numbers are generated (use the right arrow key to scroll through the numbers). The numbers correspond to the numbered states (for this example: {47 21 9 23 51 13 25 4}. If any numbers are repeated, generate a different number by using 5:randInt(51,1)). Here, the states (and Washington DC) are {Arkansas, Washington DC, Idaho, Maryland, Michigan, Mississippi, Virginia, Wyoming}.

Corresponding percentages are {30.1, 22.2, 26.5, 27.1, 30.9, 34.0, 26.0, 25.1}.

Homework from 2.2

Suppose that three book publishers were interested in the number of fiction paperbacks adult consumers purchase per month. Each publisher conducted a survey. In the survey, adult consumers were asked the number of fiction paperbacks they had purchased the previous month. The results are as follows:

• Find the relative frequencies for each survey. Write them in the charts.
• Using either a graphing calculator, computer, or by hand, use the frequency column to construct a histogram for each publisher’s survey. For Publishers A and B, make bar widths of one. For Publisher C, make bar widths of two.
• In complete sentences, give two reasons why the graphs for Publishers A and B are not identical.
• Would you have expected the graph for Publisher C to look like the other two graphs? Why or why not?
• Make new histograms for Publisher A and Publisher B. This time, make bar widths of two.
• Now, compare the graph for Publisher C to the new graphs for Publishers A and B. Are the graphs more similar or more different? Explain your answer.

Often, cruise ships conduct all on-board transactions, with the exception of gambling, on a cashless basis. At the end of the cruise, guests pay one bill that covers all onboard transactions. Suppose that 60 single travelers and 70 couples were surveyed as to their on-board bills for a seven-day cruise from Los Angeles to the Mexican Riviera. Following is a summary of the bills for each group.

• Fill in the relative frequency for each group.
• Construct a histogram for the singles group. Scale the x -axis by 💲50 widths. Use relative frequency on the y -axis.
• Construct a histogram for the couples group. Scale the x -axis by 💲50 widths. Use relative frequency on the y -axis.
• List two similarities between the graphs.
• List two differences between the graphs.
• Overall, are the graphs more similar or different?
• Construct a new graph for the couples by hand. Since each couple is paying for two individuals, instead of scaling the x -axis by 💲50, scale it by 💲100. Use relative frequency on the y -axis.
• How did scaling the couples graph differently change the way you compared it to the singles graph?
• Based on the graphs, do you think that individuals spend the same amount, more or less, as singles as they do person by person as a couple? Explain why in one or two complete sentences.
• See [link] and [link] .

• Both graphs have a single peak.
• Both graphs use class intervals with width equal to 💲50.
• The couples graph has a class interval with no values.
• It takes almost twice as many class intervals to display the data for couples.
• Answers may vary. Possible answers include: The graphs are more similar than different because the overall patterns for the graphs are the same.
• Check student’s solution.
• Both graphs display 6 class intervals.
• Both graphs show the same general pattern.
• Answers may vary. Possible answers include: Although the width of the class intervals for couples is double that of the class intervals for singles, the graphs are more similar than they are different.
• Answers may vary. Possible answers include: You are able to compare the graphs interval by interval. It is easier to compare the overall patterns with the new scale on the Couples graph. Because a couple represents two individuals, the new scale leads to a more accurate comparison.
• Answers may vary. Possible answers include: Based on the histograms, it seems that spending does not vary much from singles to individuals who are part of a couple. The overall patterns are the same. The range of spending for couples is approximately double the range for individuals.

Twenty-five randomly selected students were asked the number of movies they watched the previous week. The results are as follows.

• Construct a histogram of the data.
• Complete the columns of the chart.

Use the following information to answer the next two exercises: Suppose one hundred eleven people who shopped in a special T-shirt store were asked the number of T-shirts they own costing more than 💲19 each.

The percentage of people who own at most three T-shirts costing more than 💲19 each is approximately:

• Cannot be determined

If the data were collected by asking the first 111 people who entered the store, then the type of sampling is:

• simple random
• convenience

Following are the 2010 obesity rates by U.S. states and Washington, DC.

Construct a bar graph of obesity rates of your state and the four states closest to your state. Hint: Label the x -axis with the states.

Answers will vary.

Homework from 2.3

The median age for U.S. blacks currently is 30.9 years; for U.S. whites it is 42.3 years.

Six hundred adult Americans were asked by telephone poll, “What do you think constitutes a middle-class income?” The results are in [link] . Also, include left endpoint, but not the right endpoint.

• What percentage of the survey answered “not sure”?
• What percentage think that middle-class is from 💲25,000 to 💲50,000?
• Should all bars have the same width, based on the data? Why or why not?
• How should the <20,000 and the 100,000+ intervals be handled? Why?
• Find the 40 th and 80 th percentiles
• Construct a bar graph of the data
• 1 – (0.02+0.09+0.19+0.26+0.18+0.17+0.02+0.01) = 0.06
• 0.19+0.26+0.18 = 0.63

40 th percentile will fall between 30,000 and 40,000

80 th percentile will fall between 50,000 and 75,000

Given the following box plot:

• which quarter has the smallest spread of data? What is that spread?
• which quarter has the largest spread of data? What is that spread?
• find the interquartile range ( IQR ).
• are there more data in the interval 5–10 or in the interval 10–13? How do you know this?
• need more information

The following box plot shows the U.S. population for 1990, the latest available year.

• Are there fewer or more children (age 17 and under) than senior citizens (age 65 and over)? How do you know?
• 12.6% are age 65 and over. Approximately what percentage of the population are working age adults (above age 17 to age 65)?
• more children; the left whisker shows that 25% of the population are children 17 and younger. The right whisker shows that 25% of the population are adults 50 and older, so adults 65 and over represent less than 25%.

Homework from 2.4

In a survey of 20-year-olds in China, Germany, and the United States, people were asked the number of foreign countries they had visited in their lifetime. The following box plots display the results.

• In complete sentences, describe what the shape of each box plot implies about the distribution of the data collected.
• Have more Americans or more Germans surveyed been to over eight foreign countries?
• Compare the three box plots. What do they imply about the foreign travel of 20-year-old residents of the three countries when compared to each other?

Given the following box plot, answer the questions.

• Think of an example (in words) where the data might fit into the above box plot. In 2–5 sentences, write down the example.
• What does it mean to have the first and second quartiles so close together, while the second to third quartiles are far apart?
• Answers will vary. Possible answer: State University conducted a survey to see how involved its students are in community service. The box plot shows the number of community service hours logged by participants over the past year.
• Because the first and second quartiles are close, the data in this quarter is very similar. There is not much variation in the values. The data in the third quarter is much more variable or spread out. This is clear because the second quartile is so far away from the third quartile.

Given the following box plots, answer the questions.

• Data 1 has more data values above two than Data 2 has above two.
• The data sets cannot have the same mode.
• For Data 1 , there are more data values below four than there are above four.
• For which group, Data 1 or Data 2, is the value of “7” more likely to be an outlier? Explain why in complete sentences.

A survey was conducted of 130 purchasers of new BMW 3 series cars, 130 purchasers of new BMW 5 series cars, and 130 purchasers of new BMW 7 series cars. In it, people were asked the age they were when they purchased their car. The following box plots display the results.

• In complete sentences, describe what the shape of each box plot implies about the distribution of the data collected for that car series.
• Which group is most likely to have an outlier? Explain how you determined that.
• Compare the three box plots. What do they imply about the age of purchasing a BMW from the series when compared to each other?
• Look at the BMW 5 series. Which quarter has the smallest spread of data? What is the spread?
• Look at the BMW 5 series. Which quarter has the largest spread of data? What is the spread?
• Look at the BMW 5 series. Estimate the interquartile range (IQR).
• Look at the BMW 5 series. Are there more data in the interval 31 to 38 or in the interval 45 to 55? How do you know this?
• Each box plot is spread out more in the greater values. Each plot is skewed to the right, so the ages of the top 50% of buyers are more variable than the ages of the lower 50%.
• The BMW 3 series is most likely to have an outlier. It has the longest whisker.
• Comparing the median ages, younger people tend to buy the BMW 3 series, while older people tend to buy the BMW 7 series. However, this is not a rule, because there is so much variability in each data set.
• The second quarter has the smallest spread. There seems to be only a three-year difference between the first quartile and the median.
• The third quarter has the largest spread. There seems to be approximately a 14-year difference between the median and the third quartile.
• IQR ~ 17 years
• There is not enough information to tell. Each interval lies within a quarter, so we cannot tell exactly where the data in that quarter is concentrated.
• The interval from 31 to 35 years has the fewest data values. Twenty-five percent of the values fall in the interval 38 to 41, and 25% fall between 41 and 64. Since 25% of values fall between 31 and 38, we know that fewer than 25% fall between 31 and 35.

Twenty-five randomly selected students were asked the number of movies they watched the previous week. The results are as follows:

Construct a box plot of the data.

Homework from 2.5

The most obese countries in the world have obesity rates that range from 11.4% to 74.6%. This data is summarized in the following table.

• What is the best estimate of the average obesity percentage for these countries?
• The United States has an average obesity rate of 33.9%. Is this rate above average or below?
• How does the United States compare to other countries?

[link] gives the percent of children under five considered to be underweight. What is the best estimate for the mean percentage of underweight children?

The mean percentage, [latex]\overline{x}=\frac{1328.65}{50}=26.75[/latex]

Homework from 2.6

The median age of the U.S. population in 1980 was 30.0 years. In 1991, the median age was 33.1 years.

• What does it mean for the median age to rise?
• Give two reasons why the median age could rise.
• For the median age to rise, is the actual number of children less in 1991 than it was in 1980? Why or why not?

Homework from 2.7

Use the following information to answer the next nine exercises: The population parameters below describe the full-time equivalent number of students (FTES) each year at Lake Tahoe Community College from 1976–1977 through 2004–2005.

• μ = 1000 FTES
• median = 1,014 FTES
• σ = 474 FTES
• first quartile = 528.5 FTES
• third quartile = 1,447.5 FTES
• n = 29 years

A sample of 11 years is taken. About how many are expected to have a FTES of 1014 or above? Explain how you determined your answer.

The median value is the middle value in the ordered list of data values. The median value of a set of 11 will be the 6th number in order. Six years will have totals at or below the median.

75% of all years have an FTES:

The population standard deviation = _____

What percentage of the FTES was from 528.5 to 1447.5? How do you know?

What is the IQR ? What does the IQR represent?

How many standard deviations away from the mean is the median?

Additional Information: The population FTES for 2005–2006 through 2010–2011 was given in an updated report. The data are reported here.

Calculate the mean, median, standard deviation, the first quartile, the third quartile and the IQR . Round to one decimal place.

• mean = 1,809.3
• median = 1,812.5
• standard deviation = 151.2
• first quartile = 1,690
• third quartile = 1,935

Construct a box plot for the FTES for 2005–2006 through 2010–2011 and a box plot for the FTES for 1976–1977 through 2004–2005.

Compare the IQR for the FTES for 1976–77 through 2004–2005 with the IQR for the FTES for 2005-2006 through 2010–2011. Why do you suppose the IQR s are so different?

Hint: Think about the number of years covered by each time period and what happened to higher education during those periods.

Three students were applying to the same graduate school. They came from schools with different grading systems. Which student had the best GPA when compared to other students at his school? Explain how you determined your answer.

A music school has budgeted to purchase three musical instruments. They plan to purchase a piano costing 💲3,000, a guitar costing 💲550, and a drum set costing 💲600. The mean cost for a piano is 💲4,000 with a standard deviation of 💲2,500. The mean cost for a guitar is 💲500 with a standard deviation of 💲200. The mean cost for drums is 💲700 with a standard deviation of 💲100. Which cost is the lowest, when compared to other instruments of the same type? Which cost is the highest when compared to other instruments of the same type. Justify your answer.

For pianos, the cost of the piano is 0.4 standard deviations BELOW the mean. For guitars, the cost of the guitar is 0.25 standard deviations ABOVE the mean. For drums, the cost of the drum set is 1.0 standard deviations BELOW the mean. Of the three, the drums cost the lowest in comparison to the cost of other instruments of the same type. The guitar costs the most in comparison to the cost of other instruments of the same type.

An elementary school class ran one mile with a mean of 11 minutes and a standard deviation of three minutes. Rachel, a student in the class, ran one mile in eight minutes. A junior high school class ran one mile with a mean of nine minutes and a standard deviation of two minutes. Kenji, a student in the class, ran 1 mile in 8.5 minutes. A high school class ran one mile with a mean of seven minutes and a standard deviation of four minutes. Nedda, a student in the class, ran one mile in eight minutes.

• Why is Kenji considered a better runner than Nedda, even though Nedda ran faster than him?
• Who is the fastest runner with respect to his or her class? Explain why.

The most obese countries in the world have obesity rates that range from 11.4% to 74.6%. This data is summarized in Table 14 .

What is the best estimate of the average obesity percentage for these countries? What is the standard deviation for the listed obesity rates? The United States has an average obesity rate of 33.9%. Is this rate above average or below? How “unusual” is the United States’ obesity rate compared to the average rate? Explain.

• [latex]\overline{x}=23.32[/latex]
• Using the TI 83/84, we obtain a standard deviation of: [latex]{s}_{x}=12.95.[/latex]
• The obesity rate of the United States is 10.58% higher than the average obesity rate.
• Since the standard deviation is 12.95, we see that 23.32 + 12.95 = 36.27 is the obesity percentage that is one standard deviation from the mean. The United States obesity rate is slightly less than one standard deviation from the mean. Therefore, we can assume that the United States, while 34% obese, does not have an unusually high percentage of obese people.

[link] gives the percent of children under five considered to be underweight.

What is the best estimate for the mean percentage of underweight children? What is the standard deviation? Which interval(s) could be considered unusual? Explain.

used to describe data that is not symmetrical; when the right side of a graph looks “chopped off” compared to the left side, we say it is “skewed to the left.” When the left side of the graph looks “chopped off” compared to the right side, we say the data is “skewed to the right.” Alternatively: when the lower values of the data are more spread out, we say the data are skewed to the left. When the greater values are more spread out, the data are skewed to the right.

Introductory Statistics Copyright © 2024 by LOUIS: The Louisiana Library Network is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License , except where otherwise noted.

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New England Patriots GM would ‘be comfortable with’ any of top QBs in NFL Draft

FOXBORO, Mass. — One week before the 2024 NFL Draft , scouting director and de facto general manager Eliot Wolf spent nearly 20 minutes behind a microphone at Gillette Stadium addressing the most pressing issues and questions facing the New England Patriots . Here’s what we learned from Wolf:

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1. The Pats think they’re set up to support a rookie QB

Wolf said he’s heard the criticisms of the team’s roster. He is also familiar with the notion that New England should trade back from No. 3 for more picks to bolster the rest of the roster before adding a rookie quarterback. The idea is that a young QB might not be set up to succeed without sufficient weapons around him.

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But Wolf doesn’t see it that way. He said the public is underestimating what a rookie quarterback would have to work with on the Patriots.

“We have NFL receivers, we have NFL tight ends, we have NFL running backs, we have NFL offensive linemen,” Wolf said. “We feel good about where we are, and we feel through free agency — on the offensive side, in particular — that we’ve been able to supplement our roster properly so we’re not having to draft for need as much offensively.”

Wolf added that the Patriots have had preliminary talks with other teams about potentially acquiring wide receivers via trade but stressed New England has a good enough setup to drop a rookie quarterback into.

“We’re going to have the support that’s needed for that quarterback — if we draft one,” Wolf said. “We’re going to have a situation where they’re supported in every way possible to make them succeed.”

QB and then what? Ranking the Patriots' positions of need heading into the NFL Draft

2. They feel good about the QBs at No. 3

While Wolf said the Patriots don’t feel they have to take a quarterback at No. 3, he said they’ll “be comfortable with” any of the top options that fall to them there, presumably Jayden Daniels or Drake Maye.

That’s not a shock. We’ve known for months that the Patriots want a quarterback at No. 3. But it’s still noteworthy that he said it on the record.

Wolf added that it’s a “unique” year with six quarterbacks graded near the top of the draft.

3. They’re doing their homework

As the Patriots have done their research on the top quarterbacks, Wolf has tried to gauge how each player is viewed by his teammates. During interviews at the combine and in prospect visits to Gillette Stadium this month, they have been asking the college teammates of Daniels, Maye and J.J. McCarthy for insight.

“The best thing is hearing what their teammates say about them,” Wolf said.

LIVE: Eliot Wolf Press Conference 4/18: https://t.co/o7IYs7LLew — New England Patriots (@Patriots) April 18, 2024

4. Age doesn’t matter

There are quite a few differences between Daniels and Maye, the two quarterbacks the Patriots are most likely considering with the third pick. Their play style is different, their age is vastly different and their size is drastically different.

But one of those doesn’t matter all that much to the Patriots. Even though Daniels is one of the older quarterback prospects in the draft (he’ll turn 24 as a rookie) and Maye is one of the youngest (he’s 21), that doesn’t seem like a meaningful factor to the Patriots.

“We don’t really look at the age specifically,” Wolf said. “More the skill set.”

Dane Brugler's 7-round 2024 NFL mock draft: Predicting all 257 picks

5. They’re searching for consensus

Robert Kraft has made it clear that Wolf will have the final say in the draft room . He’s the point person for trade talks and general decision-making. But Wolf said if he’s alone in wanting to draft a certain player, he won’t do it.

“If I’m the only person that wants a player and everybody else in the building doesn’t want the player, I’m not crazy,” Wolf said. “We’re going to try to do what’s right.”

Wolf added that it’s important to try to have consensus from the top decision-makers on the most important calls, like how to use the No. 3 pick. He said the Patriots aren’t there yet, but he feels they’re close.

“We haven’t had that final conversation yet, but I do think there’s a general idea of how we feel about these players,” Wolf said.

6. They still need an outside WR

The Patriots have several wide receivers who are best utilized in the slot: Kendrick Bourne , JuJu Smith-Schuster , Demario Douglas and even K.J. Osborn . So look for them to seek an outside receiver on Day 2 of the draft even if, as Wolf said, they have some current options to play on the outside.

The Patriots’ struggle to draft good wide receivers in the early rounds is well documented. But Wolf came up in Green Bay’s scouting department and watched the Packers have ample success in that area. He’s hoping that will carry over in New England.

“(Longtime Packers GM) Ted Thompson was phenomenal at identifying receivers in Rounds 2 and 3,” Wolf said. “Hopefully some of that rubbed off on me.”

What is a best-case scenario for the Patriots in the draft?

7. Still ‘open for business,’ but no trades yet

Jerod Mayo raised some eyebrows last month at the league meetings when he acknowledged the Patriots would consider trading back from No. 3 if they received “a bag” for an offer. Wolf confirmed the Patriots are “open for business” but said the kind of offer needed to move back has yet to come.

8. Jacoby Brissett can be a mentor

Part of the reason the Patriots signed Jacoby Brissett, Wolf admitted, is they think Brissett would be an ideal mentor for a rookie quarterback, someone who knows the Patriots’ new offense and can be a good teacher.

“We signed Jacoby because he’s a good player,” Wolf said. “He’s a big, strong, relentless preparer in terms of his ability to take the game plan and apply it through the week to Sunday. He’s got a good arm, big and strong. We feel like if we end up drafting a quarterback high, he is someone who can support that player and will be a positive influence on them while competing with them.”

9. Okorafor is the left tackle … for now

Wolf doesn’t like using the phrase, “If the season started today,” because, well, it doesn’t — and because the Patriots still have next week’s draft to upgrade their roster. But for now, Wolf confirmed, free-agent signee Chukwuma Okorafor would be the projected starter at left tackle. Though Okorafor has only played right tackle in the NFL, Wolf said the Patriots studied his tape from when he was a left tackle at Western Michigan.

Still, look for the Patriots to draft a left tackle in the second or third round.

(Photos: Melina Myers and Bob Donnan / USA Today)

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Chad Graff is a staff writer for The Athletic covering the New England Patriots since 2022 after five years on the Minnesota Vikings beat. Graff joined The Athletic in January 2018 after covering a bit of everything for the St. Paul Pioneer Press. He won the Pro Football Writers of America’s 2022 Bob Oates Award for beat writing. He's a New Hampshire native and an adjunct professor of journalism at the University of New Hampshire. Follow Chad on Twitter @ ChadGraff

The relative frequency shows the proportion of data points that have each value. The frequency tells the number of data points that have each value.

Answers will vary. One possible histogram is shown below.

Find the midpoint for each class. These will be graphed on the x -axis. The frequency values will be graphed on the y -axis values.

• The 40 th percentile is 37 years.
• The 78 th percentile is 70 years.

Jesse graduated 37 th out of a class of 180 students. There are 180 – 37 = 143 students ranked below Jesse. There is one rank of 37.

x = 143 and y = 1. x + .5 y n x + .5 y n (100) = 143 + .5 ( 1 ) 180 143 + .5 ( 1 ) 180 (100) = 79.72. Jesse’s rank of 37 puts him at the 80 th percentile.

• For runners in a race, it is more desirable to have a high percentile for speed. A high percentile means a higher speed, which is faster.
• 40 percent of runners ran at speeds of 7.5 miles per hour or less (slower), and 60 percent of runners ran at speeds of 7.5 miles per hour or more (faster).

When waiting in line at the DMV, the 85 th percentile would be a long wait time compared to the other people waiting. 85 percent of people had shorter wait times than Mina. In this context, Mina would prefer a wait time corresponding to a lower percentile. 85 percent of people at the DMV waited 32 minutes or less. 15 percent of people at the DMV waited 32 minutes or longer.

The manufacturer and the consumer would be upset. This is a large repair cost for the damages, compared to the other cars in the sample. INTERPRETATION: 90 percent of the crash-tested cars had damage repair costs of $1,700 or less; only 10 percent had damage repair costs of $1,700 or more.

You can afford 34 percent of houses. 66 percent of the houses are too expensive for your budget. INTERPRETATION: 34 percent of houses cost $240,000 or less; 66 percent of houses cost $240,000 or more.

More than 25 percent of salespersons sell four cars in a typical week. You can see this concentration in the box plot because the first quartile is equal to the median. The top 25 percent and the bottom 25 percent are spread out evenly; the whiskers have the same length.

Mean: 16 + 17 + 19 + 20 + 20 + 21 + 23 + 24 + 25 + 25 + 25 + 26 + 26 + 27 + 27 + 27 + 28 + 29 + 30 + 32 + 33 + 33 + 34 + 35 + 37 + 39 + 40 = 738;

738 27 738 27 = 27.33

The most frequent lengths are 25 and 27, which occur three times. Mode = 25, 27

The data are symmetrical. The median is 3, and the mean is 2.85. They are close, and the mode lies close to the middle of the data, so the data are symmetrical.

The data are skewed right. The median is 87.5, and the mean is 88.2. Even though they are close, the mode lies to the left of the middle of the data, and there are many more instances of 87 than any other number, so the data are skewed right.

When the data are symmetrical, the mean and median are close or the same.

The distribution is skewed right because it looks pulled out to the right.

The mean is 4.1 and is slightly greater than the median, which is 4.

The mode and the median are the same. In this case, both 5.

The distribution is skewed left because it looks pulled out to the left.

Both the mean and the median are 6.

The mode is 12, the median is 13.5, and the mean is 15.1. The mean is the largest.

The mean tends to reflect skewing the most because it is affected the most by outliers.

sampling variability

induced variability

measurement variability

natural variability

For Fredo: z = .158  –  .166 .012 .158  –  .166 .012 = –0.67.

For Karl: z = .177  –  .189 .015 .177  –  .189 .015 = –.8.

Fredo’s z score of –.67 is higher than Karl’s z score of –.8. For batting average, higher values are better, so Fredo has a better batting average compared to his team.

• s x = ∑ f m 2 n − x ¯ 2 = 193,157.45 30 − 79.5 2 = 10.88 s x = ∑ f m 2 n − x ¯ 2 = 193,157.45 30 − 79.5 2 = 10.88
• s x = ∑ f m 2 n − x ¯ 2 = 380,945.3 101 − 60.94 2 = 7.62 s x = ∑ f m 2 n − x ¯ 2 = 380,945.3 101 − 60.94 2 = 7.62
• s x = ∑ f m 2 n − x ¯ 2 = 440,051.5 86 − 70.66 2 = 11.14 s x = ∑ f m 2 n − x ¯ 2 = 440,051.5 86 − 70.66 2 = 11.14
• Number the entries in the table 1–51 (includes Washington, DC; numbered vertically)
• Arrow over to PRB
• Press 5:randInt(
• Enter 51,1,8)

Eight numbers are generated (use the right arrow key to scroll through the numbers). The numbers correspond to the numbered states (for this example: {47 21 9 23 51 13 25 4}. If any numbers are repeated, generate a different number by using 5:randInt(51,1)). Here, the states (and Washington DC) are {Arkansas, Washington DC, Idaho, Maryland, Michigan, Mississippi, Virginia, Wyoming}.

Corresponding percents are {30.1, 22.2, 26.5, 27.1, 30.9, 34.0, 26.0, 25.1}.

• See Table 2.89 and Table 2.90 .
• Both graphs have a single peak.
• Both graphs use class intervals with width equal to $50
• The couples graph has a class interval with no values
• It takes almost twice as many class intervals to display the data for couples
• Answers may vary. Possible answers include the following. The graphs are more similar than different because the overall patterns for the graphs are the same.
• Check student's solution.
• Both graphs have a single peak
• Both graphs display six class intervals
• Both graphs show the same general pattern
• Answers may vary. Possible answers include the following. Although the width of the class intervals for couples is double that of the class intervals for singles, the graphs are more similar than they are different.
• Answers may vary. Possible answers include the following. You are able to compare the graphs interval by interval. It is easier to compare the overall patterns with the new scale on the couples graph. Because a couple represents two individuals, the new scale leads to a more accurate comparison.
• Answers may vary. Possible answers include the following. Based on the histograms, it seems that spending does not vary much from singles to individuals who are part of a couple. The overall patterns are the same. The range of spending for couples is approximately double the range for individuals.

Answers will vary.

• 1 – (.02+.09+.19+.26+.18+.17+.02+.01) = .06
• .19+.26+.18 = .63
• Check student’s solution.

40 th percentile will fall between 30,000 and 40,000

80 th percentile will fall between 50,000 and 75,000

• more children; the left whisker shows that 25 percent of the population are children 17 and younger; the right whisker shows that 25 percent of the population are adults 50 and older, so adults 65 and over represent less than 25 percent
• 62.4 percent
• Answers will vary. Possible answer: State University conducted a survey to see how involved its students are in community service. The box plot shows the number of community service hours logged by participants over the past year.
• Because the first and second quartiles are close, the data in this quarter is very similar. There is not much variation in the values. The data in the third quarter is much more variable, or spread out. This is clear because the second quartile is so far away from the third quartile.
• Each box plot is spread out more in the greater values. Each plot is skewed to the right, so the ages of the top 50 percent of buyers are more variable than the ages of the lower 50 percent.
• The black sports car is most likely to have an outlier. It has the longest whisker.
• Comparing the median ages, younger people tend to buy the black sports car, while older people tend to buy the white sports car. However, this is not a rule, because there is so much variability in each data set.
• The second quarter has the smallest spread. There seems to be only a three-year difference between the first quartile and the median.
• The third quarter has the largest spread. There seems to be approximately a 14-year difference between the median and the third quartile.
• IQR ~ 17 years
• There is not enough information to tell. Each interval lies within a quarter, so we cannot tell exactly where the data in that quarter is are concentrated.
• The interval from 31 to 35 years has the fewest data values. Twenty-five percent of the values fall in the interval 38 to 41, and 25 percent fall between 41 and 64. Since 25 percent of values fall between 31 and 38, we know that fewer than 25 percent fall between 31 and 35.

the mean percentage, x ¯ = 1,328.65 50 = 26.75 x ¯ = 1,328.65 50 = 26.75

The median value is the middle value in the ordered list of data values. The median value of a set of 11 will be the sixth number in order. Six years will have totals at or below the median.

• mean = 1,809.3
• median = 1,812.5
• standard deviation = 151.2
• first quartile = 1,690
• third quartile = 1,935

Hint: think about the number of years covered by each time period and what happened to higher education during those periods.

For pianos, the cost of the piano is .4 standard deviations BELOW the mean. For guitars, the cost of the guitar is 0.25 standard deviations ABOVE the mean. For drums, the cost of the drum set is 1.0 standard deviations BELOW the mean. Of the three, the drums cost the lowest in comparison to the cost of other instruments of the same type. The guitar costs the most in comparison to the cost of other instruments of the same type.

• x ¯ = 23.32 x ¯ = 23.32
• Using the TI 83/84, we obtain a standard deviation of: s x = 12.95. s x = 12.95.
• The obesity rate of the United States is 10.58 percent higher than the average obesity rate.
• Since the standard deviation is 12.95, we see that 23.32 + 12.95 = 36.27 is the disease percentage that is one standard deviation from the mean. The U.S. disease rate is slightly less than one standard deviation from the mean. Therefore, we can assume that the United States, although 34 percent have the disease, does not have an unusually high percentage of people with the disease.
• For graph, check student's solution.
• 49.7 percent of the community is under the age of 35
• Based on the information in the table, graph (a) most closely represents the data.
• 174, 177, 178, 184, 185, 185, 185, 185, 188, 190, 200, 205, 205, 206, 210, 210, 210, 212, 212, 215, 215, 220, 223, 228, 230, 232, 241, 241, 242, 245, 247, 250, 250, 259, 260, 260, 265, 265, 270, 272, 273, 275, 276, 278, 280, 280, 285, 285, 286, 290, 290, 295, 302
• 205.5, 272.5
• .84 standard deviations below the mean

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• Book title: Statistics
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