Stats in Practice: Elementary Probability Theory

• Part I - Multiple Choice Questions

  The video indicates which of the following is an acceptable alternative to washing your hands for 20 seconds with respect to preventing illness?

  rinsing with mouthwash that has at least 15% alcohol getting a flu shot     using hand sanitizer with at least 60% alcohol washing your hands for 10 seconds with water that exceeds 100 degrees Fahrenheit

  Correct. This is directly from the video. It says, "If you do not have access to a sink, using hand sanitizer with at least 60% alcohol can also fight germs."

  The video urges people to wash their hands to reduce the likelihood (that is, the probability) of contracting diseases. What does this imply?

  If you don't wash your hands, you will contract a disease. That is: P(contracting a disease if you don't wash your hands) = 1. If you contracted a disease, you must have not washed your hands. That is: P(washed your hands if you contracted a disease) = 0.      The probability of contracting a disease is lower if you wash your hands than if you don't wash your hands. That is: P(disease if you wash your hands) < P(disease if you don't wash your hands). If you wash your hands, you will not contract a disease. That is: P(contracting a disease if you wash your hands) = 0.

  
  Incorrect. The video does not say that washing your hands prevents disease with certainty, nor does it say that if you fail to wash your hands, you will contract a disease with certainty. In probability, an event that is certain has a probability of 1, while an event that will not occur, with certainty, has a probability of 0. This video was about reducing the likelihood of contracting diseases, so the implication is that if you wash your hands, your probability of contracting of disease is lower than if you did not wash your hands. This does not mean, however, that if you contracted a disease, you necessarily did not wash your hands.

  Suppose a student has had one illness in the last month, but her roommate has had two. With no other information about this student or her roommate, given the information in this video, what is your best guess about the student's and her roommate's hand-washing habits?

  The student does the best job of washing her hands. The student's roommate does the best job of washing her hands.     

  Correct. Your "best guess" should be the answer that you think is most likely. That is, the event with the higher probability. A probability describes the likelihood of an event occurring given all the information known about the event. In this case, all you know is that one student had one more illness than the other. The video actually says that washing hands prevents your "catching a bug" but later, when it discusses transmission of disease, it is not clear whether the benefit of hand-washing is to prevent you from getting sick, prevent you from getting someone else sick, or both. Since these students are roommates, either of these answers can constitute a "best guess." For example, you may believe that the person who does the worse job of washing her hands is the one who has more illnesses. Alternatively, you may believe that perhaps each student contracted a different illness outside of their shared room. Then, the student who did a worse job of washing her hands transferred her illness to her roommate, while the other one did not. That would mean, in this case, that the student who does the worse job and hand-washing has the fewest illnesses!

  Suppose you have already had a disease, D, and doctors tell you that you cannot contract it again. Assuming the doctors are correct, which of the following probability statements about your contracting the disease is correct, where

  P(D)

  stands for "the probability of contracting the disease"? (Hint: A probability is a value between 0 and 1, where 0 means that the event will not occur with certainty and 1 means the event will occur, with certainty.)

  P(D) = 0.5

  P(D) = 0

      

  P(D) = 1

  0 < P(D) < 1

  

  Correct. If you cannot, with certainty, contract the disease, then the probability of your contracting that disease is 0.

• Part II - Discussion Question

  The video claims that washing your hands saves more lives than any vaccine. Does this necessarily imply that the probability of contracting a disease, given you wash your hands, is lower than the probability of contracting a disease, given you are vaccinated?

  
  x

  Score: 1 out of 1

  Comment:

• Cities' populations

  CITY	POP
  ALBANY	846
  ALBUQUERQUE	486
  ATLANTA	2657
  ATLANTIC CITY	303
  BALTIMORE	2303
  BOSTON	2842
  BURLINGTON	127
  CHARLESTON	502
  CHARLOTTE	1091
  CHEYENNE	76
  CHICAGO	6199
  CINCINNATI	1438
  CLEVELAND	1851

  CITY	POP
  DENVER	1645
  DES MOINES	385
  DETROIT	4362
  DUBUQUE	91
  GALVESTON-T.C.	211
  HARRISBURG	584
  HARTFORD	748
  HOUSTON	3228
  HUNTINGTON	323
  INDIANAPOLIS	1229
  JACKSON	396
  JACKSONVILLE	878
  LOS ANGELES	8505

  CITY	POP
  LOUISVILLE	967
  MADISON	347
  MIAMI	1791
  MILWAUKEE	1389
  MINNEAPOLIS	2336
  MOBILE	438
  MONTGOMERY	297
  NASHVILLE	956
  NEW ORLEANS	1321
  NEW YORK	8529
  NORFOLK	1346
  OKLAHOMA CITY	975
  OMAHA	616

  CITY	POP
  PHILADELPHIA	4866
  PHOENIX	1960
  PORTLAND	210
  PORTLAND	1168
  RALEIGH	665
  SALT LAKE CITY	1055
  SAN FRANCISCO	1590
  SEATTLE	1796
  SPOKANE	355
  ST. LOUIS	2458
  WASHINGTON	3646
  WICHITA	475
  WILMINGTON	559

• JMP Applet

  Click here to access the JMP applet in a new window.

• (a)

  Determine the interquartile range.

  thousand

  What percent of data is represented by this range?

  %

• (b)

  What value separates the upper 97.5% of data from the lower 2.5%?

• (c)

  Which cities are considered upper outliers according to the IQR method? (Select all that apply.)

  New York Los Angeles Washington Houston Philadelphia Detroit Chicago none of the above

  
  

• (d)

  Which cities are considered lower outliers according to the IQR method? (Select all that apply.)

  Burlington Cheyenne Dubuque Portland none of the above

  
  

Statistical Lab

• Background

  The Pew Research Center conducted surveys pertaining to relationships and how relationships are affected by the Internet in 2005 and a similar follow-up survey in 2013. A portion of the surveys focused on online dating.

  

• Guidelines

  When performing hypothesis tests, always state your null and alternative hypotheses. Clearly identify which type of test you are using and speak to any assumptions you are making. Use a 0.05 significance level throughout. Be sure to write a thorough conclusion stating not only your generic conclusion regarding the null and alternative hypotheses, but also the meaning of this decision in context.

• About the Data

  You will need to download the relationships.csv dataset. This is a selection of the data from the 2013 survey, which includes the responses from 2,252 people age 18 and over. You will find five columns of data in the file. The data was collected during telephone interviews conducted by Princeton Survey Research Associates International from April 17 to May 19, 2013. Interviews were conducted in English and Spanish by landline (1,125) and cell phone (1,127). At least seven attempts were made to complete an interview at a sampled telephone number. The calls were staggered over times of day and days of the week to maximize the chances of making contact with a potential respondent. For the landline sample, interviewers asked to speak with the youngest adult male or female currently at home based on a random rotation. If no male/female was available, interviewers asked to speak with the youngest adult of the other gender. For the cellular sample, interviews were conducted with the person who answered the phone.

  • age—the age of the interviewee, where 97 = 97 or older, 98 = don't know, and 99 = refused to answer.
  • gender—the gender of the interviewee, where 1 = male, 2 = female.
  • onlineDating—response to the question: "Have YOU, personally, ever used an online dating site?", where 1 = yes, 2 = no, 8 = don't know, 9 = refused to answer. If blank, assume the question did not apply to the interviewee.
  • metOnline—response to the question: "Is your [spouse / current partner] someone you first met ONLINE or someone you first met OFFLINE?", where 1 = met online, 2 = met offline, 8 = don't know, 9 = refused to answer. If blank, assume the question did not apply to the interviewee.
  • desperate—response to the question: "Do you agree or disagree with this statement: People who use online dating sites are desperate", where 1 = agree, 2 = disagree, 8 = don't know, 9 = refused to answer.
  • goodWay—response to the question: "Do you agree or disagree with this statement: Online dating is a good way to meet people", where 1 = agree, 2 = disagree, 8 = don't know, 9 = refused to answer.

• Tech Guide

  You will need to use the Excel Tech Guide.

• Lab Questions

  In 2005, the mean age of people who reported having used an online dating site was 46 years old. Was the mean age of people who had used an online dating site different in 2013 than it was in 2005?

  State the null and alternative hypotheses.

  H₀: μ = 46

  
  

  H₁: μ ≠ 46

  H₀: p = 46

  
  

  H₁: p ≠ 46

      

  H₀: σ² = 46

  
  

  H₁: σ² ≠ 46

  H₀: μ₁ = μ₂

  
  

  H₁: μ₁ ≠ μ₂

  H₀: p₁ = p₂

  
  

  H₁: p₁ ≠ p₂

  Which test is most appropriate?

  a hypothesis test concerning a population mean when σ is known a hypothesis test concerning a population mean when σ is unknown     a hypothesis test concerning a population proportion a hypothesis test concerning a population variance or standard deviation a hypothesis test concerning the difference in population means a hypothesis test concerning the difference in population proportions

  State your conclusion regarding the null and alternative hypotheses.

  reject the null hypothesis based on a 0.05 significance level do not reject the null hypothesis based on a 0.05 significance level    

  What is your conclusion in the context of the problem?

  There is sufficient evidence to conclude that the mean age of people who had used an online dating site was different in 2013 than it was in 2005. There is insufficient evidence to conclude that the mean age of people who had used an online dating site was different in 2013 than it was in 2005.    

  In 2005, 2.6% of people who were married or in a serious relationship said they met online. Has this proportion increased in recent years?

  State the null and alternative hypotheses.

  H₀: μ = 0.026

  
  

  H₁: μ > 0.026

  H₀: p = 0.026

  
  

  H₁: p > 0.026

      

  H₀: σ² = 0.026

  
  

  H₁: σ² > 0.026

  H₀: μ₁ = μ₂

  
  

  H₁: μ₁ ≠ μ₂

  H₀: p₁ = p₂

  
  

  H₁: p₁ ≠ p₂

  Which test is most appropriate?

  a hypothesis test concerning a population mean when σ is known a hypothesis test concerning a population mean when σ is unknown     a hypothesis test concerning a population proportion a hypothesis test concerning a population variance or standard deviation a hypothesis test concerning the difference in population means a hypothesis test concerning the difference in population proportions

  State your conclusion regarding the null and alternative hypotheses.

  reject the null hypothesis based on a 0.05 significance level do not reject the null hypothesis based on a 0.05 significance level    

  What is your conclusion in the context of the problem?

  There is sufficient evidence to conclude that the number people who were married or in a serious relationship and said they met online has increased in recent years. There is insufficient evidence to conclude that the number people who were married or in a serious relationship and said they met online has increased in recent years.    

  In 2005, 29.5% of people agreed with the statement "People who use online dating are desperate." Was this proportion different in 2013?

  State the null and alternative hypotheses.

  H₀: μ = 0.295

  
  

  H₁: μ ≠ 0.295

  H₀: p = 0.295

  
  

  H₁: p ≠ 0.295

      

  H₀: σ² = 0.295

  
  

  H₁: σ² ≠ 0.295

  H₀: μ₁ = μ₂

  
  

  H₁: μ₁ ≠ μ₂

  H₀: p₁ = p₂

  
  

  H₁: p₁ ≠ p₂

  Which test is most appropriate?

  a hypothesis test concerning a population mean when σ is known a hypothesis test concerning a population mean when σ is unknown     a hypothesis test concerning a population proportion a hypothesis test concerning a population variance or standard deviation a hypothesis test concerning the difference in population means a hypothesis test concerning the difference in population proportions

  State your conclusion regarding the null and alternative hypotheses.

  reject the null hypothesis based on a 0.05 significance level do not reject the null hypothesis based on a 0.05 significance level    

  What is your conclusion in the context of the problem?

  There is sufficient evidence to conclude that the proportion of people who agreed with the statement "People who use online dating are desperate" was different in 2013. There is insufficient evidence to conclude that the proportion of people who agreed with the statement "People who use online dating are desperate" was different in 2013.    

  Based on the 2013 survey, is the mean age of those who agreed with the statement "People who use online dating are desperate" different from those who disagreed?

  State the null and alternative hypotheses.

  H₀: μ₁ = μ₂

  
  

  H₁: μ₁ ≠ μ₂

  H₀: p₁ = p₂

  
  

  H₁: p₁ ≠ p₂

      

  H₀: σ₁² = σ₂²

  
  

  H₁: σ₁² ≠ σ₂²

  H₀: μ₁ ≠ μ₂

  
  

  H₁: μ₁ = μ₂

  H₀: p₁ ≠ p₂

  
  

  H₁: p₁ = p₂

  Which test is most appropriate?

  a hypothesis test concerning a population mean when σ is known a hypothesis test concerning a population mean when σ is unknown     a hypothesis test concerning a population proportion a hypothesis test concerning a population variance or standard deviation a hypothesis test concerning the difference in population means a hypothesis test concerning the difference in population proportions

  State your conclusion regarding the null and alternative hypotheses.

  reject the null hypothesis based on a 0.05 significance level do not reject the null hypothesis based on a 0.05 significance level    

  What is your conclusion in the context of the problem?

  There is sufficient evidence to conclude that the mean ages for the two groups are different. There is insufficient evidence to conclude that the mean ages for the two groups are different.    

  Based on the 2013 survey, do young people (age 18–29) tend to agree more with the statement "Online dating is a good way to meet people" than people who are over 29?

  State the null and alternative hypotheses.

  H₀: μ₁ = μ₂

  
  

  H₁: μ₁ > μ₂

  H₀: p₁ = p₂

  
  

  H₁: p₁ > p₂

      

  H₀: σ₁² = σ₂²

  
  

  H₁: σ₁² > σ₂²

  H₀: μ₁ ≠ μ₂

  
  

  H₁: μ₁ = μ₂

  H₀: p₁ ≠ p₂

  
  

  H₁: p₁ > p₂

  Which test is most appropriate?

  a hypothesis test concerning a population mean when σ is known a hypothesis test concerning a population mean when σ is unknown     a hypothesis test concerning a population proportion a hypothesis test concerning a population variance or standard deviation a hypothesis test concerning the difference in population means a hypothesis test concerning the difference in population proportions

  State your conclusion regarding the null and alternative hypotheses.

  reject the null hypothesis based on a 0.05 significance level do not reject the null hypothesis based on a 0.05 significance level    

  What is your conclusion in the context of the problem?

  There is sufficient evidence to conclude that young people tend to agree more with the statement "Online dating is a good way to meet people." There is insufficient evidence to conclude that young people tend to agree more with the statement "Online dating is a good way to meet people."    

  Write a paragraph that weaves together your findings (and any additional tests or estimates you may wish to make) in a meaningful way.

  
  

  This answer has not been graded yet.

Milestone 1: Research Design

• Question 1

  What is your research question?

  Your research question should be based on a topic that interests you and that you can reasonably obtain data for. Try to make your research question as specific as possible. Form a research question about a population that you will be able to sample. Some examples of research questions are: "Are students at my college able to taste the difference between regular coffee and decaffeinated coffee?", "Does the GPA, age, and number of credits needed for graduation for juniors at my university differ between transfer students and non-transfer students?", "Have the new water rates in my water district changed residents' water usage habits?", and "Do recent graduates from the business department at my university get larger starting salaries on average if they have participated in a summer internship?"

  
  

  This answer has not been graded yet.

• Question 2

  Has any research already been done that might help with your project?

  You can search your university's online database to find articles that discuss research that has already been done on your research question. Please make note of any articles or other sources that helped guide your research question and research design. You will be asked to provide references to any sources you used.

  
  

  This answer has not been graded yet.

• Question 3

  What is your population of interest?

  Your population of interest will be closely related to your research question. For instance, if you are testing the freshman 15 theory and you believe that students at your college do not gain 15 pounds on average over the course of freshman year, then your population of interest is the students at your college who have completed freshman year.

  
  

  This answer has not been graded yet.

• Question 4

  Describe your sampling plan.

  You want to ensure that your sample is representative of the population of interest. How will you target your population of interest? Sampling involves tradeoffs; you may have to settle for a convenience sample, which is fine for this project as long as you acknowledge this. Sampling is about doing the best you can with the resources you have at your disposal.

  
  

  This answer has not been graded yet.

• Question 5

  Will this sample be representative of the population of interest? If so, why? If not, what population will the sample be representative of?

  Is there any group of people within the population of interest that your sampling plan is likely to miss? For instance, if you are using a random sample of landline telephone numbers from your state in order to achieve a random sample of residents in your state, then you are missing all of the residents of your state who only have a cell phone and do not have a landline. Or, if your population of interest is all adults in your country, but you only conduct interview of adults from your local community, then your sample is only representative of the adults in your local community.

  
  

  This answer has not been graded yet.

• Question 6

  What variables will you measure?

  Make sure that the variables you choose are measurable. If you are performing an experiment where you are comparing two skin creams to determine which gives the skin a more youthful glow, how exactly will you measure "youthful glow"? Are there other variables that you will collect because they too may be of interest? Age? Gender? Level of education? Is your variable type numeric or categorical? If your variable type is numeric, is it discrete or continuous? Do your variables need to be measured using special equipment?

  Variable	Type	How will you measure it?
  	---Select--- numeric - discrete numeric - continuous categorical	This answer has not been graded yet.
  	---Select--- numeric - discrete numeric - continuous categorical	This answer has not been graded yet.
  	---Select--- numeric - discrete numeric - continuous categorical	This answer has not been graded yet.
  	---Select--- numeric - discrete numeric - continuous categorical	This answer has not been graded yet.

• Question 7

  Describe your study design.

  Will you be conducting an observational study or performing an experiment? What will your sample size be? If you will be performing an experiment, will there be a placebo? If so, describe the placebo. Will the experiment be blinded? Double-blinded? Who will carry out the experiment and how?

  
  

  This answer has not been graded yet.

Milestone 2: Gather Data

Please select ONE method for gathering data.

• Method 1 - Create your own dataset.

  Create a survey:

  What are your main questions? Will the responses to your questions provide answers to your research question? Be conscious of the language you use—you don't want the phrasing of your question to influence the responses. Are there any follow-up questions? Do your questions ensure that you will have values for all of your variables? Are your questions multiple choice or free response? If some of your questions are free response, will you be able to analyze this data using the methods you have learned? Be conscious of how much time a respondent will spend answering your survey.

  Gather data:

  Use your sampling plan as a guide for collecting your data, always keeping your population of interest in mind.

  The file will be uploaded when you submit. (Submit a file with a maximum size of 1 MB.)

  This answer has not been graded yet.

• Method 2 - Use an existing dataset.

  If you choose to use existing data, you will first need to locate it, download it, and document it. Remember you will need to understand how the data was gathered.

  The following are a few sites to get you started.

  • Baseball Prospectus
  • US Census: Business and Industry
  • FedStats

  The file will be uploaded when you submit. (Submit a file with a maximum size of 1 MB.)

  This answer has not been graded yet.

Milestone 3: Analyze Data

• Question 1

  Get to know your data using the statistical software of your choosing, e.g., Excel, JMP, Minitab, R, SPSS, TI-Calculator, etc.

  Organize, summarize, and plot your data. The type of summaries and plots will depend on the type of data you have. For categorical data, you'll perhaps want to look at bar charts. For numeric data, you'll perhaps want to look at scatterplots or histograms. Look for any values that seem unusually high or low. Double check to make sure there aren't typos.

• Question 2

  Describe the characteristics of your data.

  If you have categorical data, describe the overall patterns in your data. If you have univariate numerical data, describe the shape of the distribution of your data. If you have paired data, describe the relationship between the pairs. Negative? Positive? Highly correlated?

  
  

  This answer has not been graded yet.

• Question 3

  Select the appropriate data analysis method to answer your primary research question.

  The method you use to analyze your data will depend heavily on your data. Is your data categorical or quantitative? Do you have one variable? Two variables? More than two variables? Is your data approximately normal? Do you know the population standard deviation? Do you have a large sample size? Make sure to show evidence that you checked the requirements of any statistical method that you plan to use.

  Some examples:

  • Hypothesis test for one population mean using normal distribution (or using a Student's t distribution)
  • Hypothesis test for the difference of two population means (or for the population mean difference using paired samples)
  • Hypothesis test for a population proportion (or for the difference of two population proportions)
  • Hypothesis test for a population variance (or for the equality of two population variances)
  • Goodness-of-fit test
  • Chi-square test of homogeneity (or independence)
  • One-way ANOVA
  • Tukey-Kramer (T-K) multiple comparisons procedure
  • Two-way ANOVA
  • Matched pairs sign test
  • Single-sample sign test
  • Wilcoxon Rank Sum Test
  • Kruskal-Wallis Test
  • McNemar's Test

  
  

  This answer has not been graded yet.

• Question 4

  Attach your output.

  How you analyze your data will depend heavily on the method you choose to analyze your data. We recommend you use your choice of statistical software to analyze the data. If you're getting results you don't expect, check the raw data and make sure all values are correctly entered and correctly formatted.

  The file will be uploaded when you submit. (Submit a file with a maximum size of 1 MB.)

  This answer has not been graded yet.

Milestone 4: Present Results

1. Introduction
   Introduce your research question. You might want to describe why you chose this topic. The goal of the introduction is to explain "Why should we care?"
2. Methodology
   Describe how you formulated your null and alternative hypothesis based on your research question. Describe how you collected your sample. Describe any challenges faced and how you overcame them. Describe what methods you used to analyze your data. Did you run into any challenges? Did you have to change your methods due to quirks in the data? Were there outliers in your data? If so, what did you do with them? Was any of your data from a group that was not representative of the population of interest? If so, what did you do with this data?
3. Interpretation
   Did you find a statistically significant result? What does this say about the population of interest? If it's statistically significant, is it also practically significant? Explain and consider using confidence intervals or some other measure of effect size to justify your answer. If you did not find a statistically significant result, describe why you think you didn't find one. Discuss potential errors that might have been made during the sampling and/or analysis. What are the consequences of these errors?
4. Conclusions and Future Research
   Wrap up by describing your findings in the context of your research question. What did you learn about your research question and research in general? If you were to do this project over again, what would you do differently given what you have learned? If you or another researcher were to carry out additional research on this topic, what would your advice to them be? What are some possible next steps or follow-up research questions?
5. References
   Provide references for any sources you used in the project. This includes articles you might have read or data sources you used to help you create this project.

• Written Report

  Attach a written report (.TXT, .DOC, or .DOCX files accepted) or use the template below.

  The file will be uploaded when you submit. (Submit a file with a maximum size of 1 MB.)

  This answer has not been graded yet.

  1.

  Introduction

  
  

  This answer has not been graded yet.

  2.

  Methodology

  
  

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  3.

  Interpretation

  
  

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  4.

  Conclusions and Future Research

  
  

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  5.

  References

  
  

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• Video Report

  Attach your video (.MP4, .WAV, or .WMV files will be accepted) or provide a link to a streaming or downloadable video file (YouTube, Google Drive, Dropbox, etc.).

  The file will be uploaded when you submit. (Submit a file with a maximum size of 9 MB)

  This answer has not been graded yet.

  Provide a link to a streaming or downloadable video file.

  
  

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