All Chapter Projects

Project A: Analyzing a Given Study

Space Tourism?

According to a survey conducted by the Pew Research Center, more US adults say they would not want to orbit the Earth than say they would. Among the $\mathrm{2,541}$ respondents from March 27-April 9, 2018, $42\%$ say they would definitely or probably be interested in orbiting the Earth in a spacecraft in the future, while roughly $58\%$ say they would not be interested. Among the many different viewpoints explored by the researchers, about one-half of men ($51\%$) say they would be interested in orbiting the Earth in a spacecraft, compared with one-third of women ($33\%$). Interest in being a space tourist is higher among younger generations and men overall. Also, $63\%$ of Millennials (born 1981 to 1996) say they would definitely or probably be interested in space tourism. Only $39\%$ of Gen Xers (born 1965 to 1980) and $27\%$ of those in the Baby Boomer or older generations would be interested. The $58\%$ of US adults who say they wouldn't want to orbit the Earth aboard a spacecraft believe that such a trip would be either “too expensive” ($28\%$ of those asked) or "too scary" ( $28\%$), or that their age or health wouldn't allow it ( $28\%$). Some $16\%$ of those not interested in space travel offered reasons other than the three options in the survey. The survey was part of the American Trends Panel (ATP), a nationally representative panel of randomly selected US adults living in households recruited from landline and cell phone random-digit-dial (RDD) surveys. Panelists participate monthly via selfadministered web surveys. Panelists who do not have internet access were provided with a tablet and wireless internet connection.

Source: Strauss, Mark and Kennedy, Brian. "Space tourism? Majority of Americans say they wouldn't be interested." Pew Research Center. 7 June 2018. http://www.pewresearch.org/fact-tank/2018/06/07/space-tourism-majority-of-americans-say-they-wouldnt-be-interested/ (9 Dec. 2018).

Analyze the surveys described above using the following exercises as a guide.

1. Identify the population being studied.

2. Identify the sample and the sample size. Is the sample representative of the population? Explain your answer.

3. Describe how the sample was chosen. Is there any potential bias in the sampling method? Explain your answer.

4. List the descriptive statistics given in the article.

5. What inferences does the article make from the descriptive statistics?

6. Who conducted the studies? Is there any potential researcher bias? Explain your answer.

Project B: Analyzing a Study You Find

Find a study of interest on the internet, in a newspaper or magazine, or in an educational journal. If you are reading studies in another course, an excellent choice is to use an assignment from that class. This would also be a good opportunity to begin reading journal articles from your major field of study. The periodical section of the library or library internet resources are wonderful starting places. Once you have found an article, make a copy to include with your analysis.

Note: Though it may be tempting to choose a very short article, if the article is too short, it might not contain all of the necessary elements.

Once you have found a study of interest, analyze it using the following exercises as a guide. Write a formal summary of your analysis.

1. Who conducted the study, when and where?

2. What question(s) does the study seek to answer?

3. Identify the population and variables being studied.

4. Identify the sample and the approximate sample size. Is the sample representative of the population? Explain your answer.

5. How do the researchers deal with the issues of confidentiality and informed consent?

6. Describe how the sample was chosen. Is there any potential bias in the sampling method? Explain your answer.

7. List the descriptive statistics given in the article.

8. Are there any confounding variables?

9. What inferences do the researchers draw from the descriptive statistics?

10. Is there any potential researcher bias?

11. Do you feel comfortable believing the results of the study based on your analysis? Explain your answer.

Fast Food Comparison

For three fast-food restaurants, a random sample of service times (in seconds) was taken. The results are below.

1. For each restaurant's service times, do the following.

   1. Create a frequency distribution with the given classes. Include the frequency, class boundaries, relative frequency, and cumulative frequency of each class.

      Class
      $30$–$49$
      $50$–$69$
      $70$–$89$
      $90$–$109$
      $110$–$129$

   2. Use the frequency distribution to construct a histogram of the data.

2. Based on the graphical descriptions of the data you created in part 1, which fast-food restaurant do you believe consistently has the shortest service times? Explain why.

Project A: Olympic Gold

The following table gives the years of the Winter Olympics and the number of gold medals won in each year by the United States through 2014.

Source: Olympic.org. “Olympic Games.” http://www.olympic.org/olympic-games (4 April 2019).

Analyze the data in the table by calculating descriptive statistics for the USA gold medal count. We will use these statistics to estimate the expected number of gold medals for the United States in the 2018 Winter Games. Note that the number of events is not the same for every year. Therefore, it will be necessary to take into account the number of gold medals as a percentage of the number of events.

1. Begin by calculating the number of gold medals won each year as a percentage of the number of events. To do this, divide the number of gold medals by the number of events, and then multiply by $100$. Round your answers to the nearest whole percentage. Create a table similar to the one above with the percentage of gold medals won in each year.

2. Calculate the range of the number of gold medals and the range of the percentage of gold medals.

3. Calculate the median number of gold medals and the median percentage of gold medals.

4. Calculate the mode of the number of gold medals and the mode of the percentage of gold medals.

5. Calculate the mean number of gold medals won.

6. Calculate the mean percentage of gold medals won by adding to get the total number of gold medals won in all years, dividing by the total number of events in all years and then multiplying by $100$. Round your answer to the nearest whole percentage. (Note that you cannot average percentages. This is why we have to go back to the numbers of gold medals and the numbers of events.)

7. Calculate the five-number summary for the number of gold medals won.

8. Draw a box plot for the number of gold medals won.

9. Using a value of $102$ events in the 2018 Olympics, estimate the expected number of gold medals by multiplying $102$ by the median percentage of gold medals won by the United States.

10. The United States actually won $9$ gold medals in the 2018 Winter Games in Pyeongchang, South Korea. How does your calculation for the expected number of gold medals compare to the actual number?

Project B: Where Would You Invest Your Money?

Let's look at several sets of stock prices. The following prices were obtained from historical records from 2019, and are listed in dollars per share.

Source: Yahoo! Finance. http://finance.yahoo.com (4 April. 2019).

1. Find the mean of each set of stock prices.

2. Find the median of each set of stock prices.

3. Calculate the variance and standard deviation of each set of stock prices.

4. If you have $\mathrm{\$}\mathrm{10,000}$ to invest, what stock would you buy under the following circumstances? Justify your reasoning.

   1. You are nearing retirement and need a stable investment for the future.

   2. You are a wealthy entrepreneur hoping to make a large profit in a short amount of time.

Law of Large Numbers

This project is designed to be completed in groups, but it may be completed by an individual. Each group is to follow the steps below using a standard sixsided die.

Playing Roulette

We all dream of winning big, becoming an instant millionaire; but how likely is that? Let's say we decide to pursue our goal of winning big money by going to a casino and continually playing what we think will be an easy game: roulette. Can we expect to win big in the long run? Is one bet better than another? How do the casinos make so much money anyway? If we are betting against the casino, how do they make sure that they always win? This project will help you answer these questions.

Let's begin with a lesson in roulette. Roulette is a casino game that involves spinning a ball on a wheel that is marked with numbered squares that are red, black, or green. Half of the numbers $1$–$36$ are colored red and half are black and the numbers $0$ and $00$ are green. Each number occurs only once on the wheel.

We can make many different types of bets, but two of the most common are to bet on a single number ($1$–$36$) or to bet on a color (either red or black). These will be the two bets we will consider in this project. After all players place their bets on the table, the wheel is spun and the ball tossed onto the wheel. The pocket in which the ball lands on the wheel determines the winning number and color. The ball can land on only one color and number at a time.

We begin by placing a bet on a number between $1$ and $36$. This bet pays $36$ to $1$ in most casinos, which means we will be paid $\mathrm{\$}36$ for each $\mathrm{\$}1$ we bet on the winning number. If we lose, we simply lose whatever amount of money we bet.

1. Calculate the probability that we will win on a single spin of the wheel.

2. Calculate the probability that we will lose.

3. If we bet $\mathrm{\$}8$ on the winning number, how much money will we win?

4. What is the expected value of a bet on a single number if we bet $\mathrm{\$}1$?

5. For a $\mathrm{\$}5$ bet, what is the expected value of a bet on a single number?

6. What is the expected value of a bet on a single number if we bet $\mathrm{\$}10$?

7. Do you see a pattern in the answers to the last three questions?

We decide that we can certainly increase our chances of winning if we bet on a color instead of a number. Roulette allows us to bet on either red or black and if the number is that color, we win. This bet pays even money in most casinos. This means that for each dollar we bet, we will win $\mathrm{\$}1$ for choosing the winning color. So, if we bet $\mathrm{\$}5$ and win, we would keep our $\mathrm{\$}5$ and win $\mathrm{\$}5$ more. If we lose, we lose whatever amount of money we bet, just as before.

8. What is the probability that we will win on a single spin if we bet on red?

9. What is the probability that we will lose on a single spin if we bet on red?

10. If we bet $\mathrm{\$}60$ on the winning color, how much money will we win? Is this more or less than we will win by betting $\mathrm{\$}8$ on our favorite number? Explain why.

11. What is the expected value of a bet on red if we bet $\mathrm{\$}1$?

12. For a $\mathrm{\$}5$ bet, what is the expected value of a bet on red?

13. What is the expected value of a bet on red if we bet $\mathrm{\$}10$?

14. Do you see a pattern in the answers to the last three questions?

15. How does the expected value of betting on a number compare to the expected value of betting on a color? Is one bet more profitable than another?

16. If our goal was to play roulette so that we can “win it big,” what does the expected value of a bet tell us about our chances of winning a large amount of money?

17. Are the casinos really gambling when we place a bet against them? Explain.

Curving Grades Using a Normal Distribution

Dr. Smith, a biology professor at Bradford University, has decided to give his classes a standardized biology exam that is nationally normed. This indicates that the normal distribution is an appropriate approximation for the probability distribution of students' scores on this exam. The probability distribution of students' scores on this standardized exam can be estimated using the normal distribution shown below.

1. State the mean of the distribution of the biology exam scores.

2. State the standard deviation of the distribution of the biology exam scores.

Grading Curve Option I

Originally, Dr. Smith decides to curve his students' exam grades as follows.

• Students whose scores are at or above the ${90}^{\text{th}}$ percentile will receive an A.

• Students whose scores are in the ${80}^{\text{th}}$–${89}^{\text{th}}$ percentiles will receive a B.

• Students whose scores are in the ${70}^{\text{th}}$–${79}^{\text{th}}$ percentiles will receive a C.

• Students whose scores are in the ${60}^{\text{th}}$–${69}^{\text{th}}$ percentiles will receive a D.

• Students whose scores are below the ${60}^{\text{th}}$ percentile will receive an F.

3. Find the z-scores that correspond to the following percentiles.

   • ${90}^{\text{th}}$ percentile

   • ${80}^{\text{th}}$ percentile

   • ${70}^{\text{th}}$ percentile

   • ${60}^{\text{th}}$ percentile

4. Using that information, find the exam scores that correspond to the curved grading scale. Assume that the exam scores range from 0 to 100. (Round to the nearest whole number.)

   A:     –$100$

   B:     –    

   C:     –    

   D:     –    

   F: $\mathrm{0–}$    

5. The following is a partial list of grades for students in Dr. Smith's class. Using the grading scale you just created, find the new curved letter grades that the students will receive on their tests given their raw scores.

   Biology Exam Grades

   Name	Raw Score	Grade
   Adam	$82$	B
   Bill	$77$	C
   Susie	$91$	A
   Troy	$86$	B
   Sharon	$75$	C
   Laura	$66$	D
   Eric	$88$	B
   Marcus	$69$	D
   Stephanie	$79$	C

Grading Curve Option II

After reviewing the results, Dr. Smith decides to consider an alternate curving method. He decides to assign exam grades as follows.

• A: Students whose scores are at least two standard deviations above the mean of the standardized test.

• B: Students whose scores are from one up to two standard deviations above the mean of the standardized test.

• C: Students whose scores are from one standard deviation below the mean up to one standard deviation above the mean of the standardized test.

• D: Students whose scores are from two standard deviations below the mean up to one standard deviation below the mean of the standardized test.

• F: Students whose scores are more than two standard deviations below the mean of the standardized test.

6. Using the previous information, create Dr. Smith's new grading scale. (Round to the nearest whole number.)

   A:     –$100$

   B:     –    

   C:     –    

   D:     –    

   F: $\mathrm{0–}$    

7. Using the grading scale you just created, return to the partial list of grades and find the new curved letter grades that the students will receive on their tests given their raw scores.

8. Review the grades each student received using the two grading scales. Which grading scale do you feel is fairer? Explain why.

Project A: Central Limit Theorem Experiment

You will need a standard six-sided die and at least six sets of data to complete this project.

Consider the distribution of the possible outcomes from rolling a single die; that is, $1$, $2$, $3$, $4$, $5$, and $6$. Let's use this distribution as our theoretical population distribution. We want to use this population distribution to explore the properties of the Central Limit Theorem. Let's begin by determining the shape, center, and dispersion of the population distribution.

1. What would you expect the distribution of the outcomes from repeated rolls of a single die to look like; in other words, what is its shape? (Hint: What is the probability of getting each value?)

   Shape:     

2. Calculate the mean of the population. (Hint: What is the mean outcome for rolling a single die?)

   $\mu =\underline{\phantom{000000000}}$

3. Calculate the standard deviation of the population. (Hint: What is the standard deviation of all possible outcomes from rolling a single die?)

   $\sigma =\underline{\phantom{000000000}}$

Let's continue by exploring the distribution of the original population empirically. To do so, follow these steps.

The Central Limit Theorem is not about individual rolls like we just looked at, but is about the averages of sample rolls. Thus we need to create samples in order to explore the properties of the Central Limit Theorem.

The Central Limit Theorem says that the distribution of the sample means should be closer to a normal distribution when the sample size becomes larger. To see this effect, group your original data from Step 1 into two samples of $30$ rolls instead of six sets of $10$. Repeat Steps 5–11 using the new sample size of $n=30$.

Project A: Constructing a Confidence Interval for a Population Mean

Choose a study question and follow the given steps to construct a confidence interval for a population mean.

Choose one of the study questions below, or write your own study question for which you could collect data from students on your campus to estimate a population mean.

• How many hours per week do college students on your campus study?

• What is the mean number of parking tickets college students on your campus receive per semester?

• What is the mean price of rent paid per month by college students on your campus?

For the study question you chose, construct a confidence interval for that parameter by doing each of the following.

1. Collect data on your study question from at least $30$ students on your campus.

2. Calculate the sample mean and sample standard deviation of your sample data.

3. Calculate the margin of error for a $95\%$ confidence interval using your sample statistics.

4. Construct a $95\%$ confidence interval using your sample statistics.

5. Write a summary of your results.

Project A: Screentime Challenge

Students sometimes get a bad reputation for the amount of time they spend on their smartphones. But what about the screen times of faculty and staff? Do you think students really have higher screen time averages than the faculty and staff at their schools? This project will allow you to investigate this claim for yourself.

Project A: Hypothesis Testing for Population Means

Choose one of the three claims, collect data from members of the appropriate population, and perform a hypothesis test to determine if the evidence supports the claim or does not support the claim. After you have written your conclusion, look at the “real” value of the population mean (given below) and determine if your hypothesis test produced a correct decision, a Type I error, or a Type II error.

Pick one of the following claims to test:

Claim 1: Parents of college freshmen believe that freshmen spend a mean of at most $\mathrm{\$}50$ per week on eating out.

Claim 2: It is believed that college sophomores see at least $4$ movies per month at the theater.

Claim 3: The librarian claims that college juniors visit the library twice a week.

Types of Errors

Let's assume we find out that the truths are as follows.

• Freshmen spend a mean of $\mathrm{\$}40$ per week eating out.

• The mean number of movies seen by sophomores at the theater each month is more than $4$.

• The mean number of times per week that juniors visit the library is $4$.

Based on your conclusion, did you make a Type I error, a Type II error, or a correct decision? Explain.

Project A: Hypothesis Testing for Two Population Parameters

Choose one of the three questions to answer, collect data from members of the appropriate population, and perform a hypothesis test to answer the question. After you have written your conclusion, look at the "truth" and determine if your hypothesis test produced a correct decision, a Type I error, or a Type II error.

Pick one of the following questions to test:

• Is there a difference in study habits for college students who are $25$ and older versus those students who are aged $18$-$24$? Ask at least $30$ students $25$ and older and at least $30$ students aged $18$-$24$ to estimate the amount of time they spend studying each week. Assume that the population variances are equal. Use a $0.10$ level of significance.

• Do college freshmen spend less money each week eating out than seniors? Ask between $10$ and $20$ freshmen and between $10$ and $20$ seniors to estimate the amount of money they spend each week eating out. Assume that the population variances are different and both population distributions are approximately normal. Use a $0.05$ level of significance.

• Are the percentages of students and faculty who exercise regularly the same? Ask at least $30$ students and at least $30$ faculty if they exercise at least three times per week. Record the number of students and the number of faculty who say "yes." Use a $0.01$ level of significance.

Types of Errors

Suppose the "truths" are as follows.

• The mean amount of time spent studying each week is higher for students 25 and older in college than for college students aged 18-24.

• The mean amount of money spent eating out each week is higher for freshmen than for seniors.

• The percentage of students who exercise at least three times per week is the same as the percentage of faculty.

Based on your conclusion, did you make a Type I error, Type II error, or a correct decision? Explain.

Arm Span vs. Height

Is it true that a person's arm span is equal to the person's height? If so, then there should be a perfect linear relationship between arm span and height. Let's find out if this is true by collecting data and creating a linear regression model.