Project Part II: Data and Descriptive Statistics In this section you will expand your project to include your actual data set, descriptive statistics for your quantitative variables, linear regression between those two variables, and a discussion of your results. Your project should be submitted as a professional report including everything from Part I and II using the following headings: Research Proposal: Introduction, Methods, and Materials from Part I Data: You must collect your own data set using your survey and following the plan you set out in part I. Include your actual data, neatly arrange in a table, one row per case with a column for each variable. Remember, you should have at least 20 cases (subjects)! For example: Case: Hours studied per week GPA Student 1* 5 3.0 Student 2 10 3.2 Student 3 8 2.9 *Note: Do not report subject names due to confidentiality Distribution of Quantitative Data: Create both a histogram and modified boxplot for EACH of your quantitative variables (this means you should have a total of 2 histograms and 2 modified boxplots). You can choose what technology to use, but it must look professional (hand drawn histograms or boxplots will receive no credit). Indicate how your graph was created using the technology (i.e. indicate what technology was used and give instructions). For each boxplot report the fences. You must show your computations for anything computed by hand and show instructions for anything computed using technology. For each quantitative variable: Use the histograms and boxplots to describe the distribution. Discuss the shape (modality, symmetry/skew, and unusual features such as gaps or outliers) of your data for each variable. For example were any of your data points designated outliers based on the fences (which are based on the 1.5 IQR rule)? Report the summary statistics (Mean, SD, 5number summary, IQR). Indicate whether the mean or median is a better measure of center, explain why, and give that value. Indicate whether the standard deviation or IQR is a better measure of spread, explain why, and give that value. Z-Scores and the Normal Distribution: Identify which one of your quantitative variables has a distribution that is closest to a normal distribution (unimodal and symmetric). Discuss which variable you chose and why. For this variable, select the highest and lowest value within your data set and compute the z-score. You must show all your calculations typed neatly using appropriate word processing software with a mathematics package (for example Equation Editor or MathType in MS Word). Answer the question: which z-score is more extreme? Also, pick one other data value from your data set. Compute its z-score. Use the normal model to approximate the probability (percent of the data) of being more extreme than this value. For example, if your z-score is negative such as -0.4 you'd want to find the probability of having a z-score LESS than -0.4. If your z-score is positive such as 1.2, you'd want to find the probability of having a z-score GREATER than 1.2. If you use technology to assist you, explain how. Linear Regression and Correlation Recall that in your research proposal you discussed that you believed there might be an association between your two quantitative variables. You are now going to examine this relationship using Linear Regression. Based on what you wrote in Part I of your project, state which variable you are selecting to be your explanatory (x) variable and which variable you are selecting to be your response (y) variable and explain why you made this decision. Scatterplot: Create a scatterplot of your explanatory and response variable. You can choose what technology to use, but it must look professional (hand drawn scatterplots will receive no credit). Based on your scatterplot, discuss the direction, form, and strength of the association using appropriate statistical terminology. Are there any suspected outliers or clusters? Is linear correlation and regression appropriate based on your scatterplot? Correlation Coefficient and Linear Regression: Report the correlation coefficient and linear regression equation using your choice of technology. Indicate how your correlation coefficient and linear regression equation were created using the technology (i.e. give detailed instructions). Does your correlation coefficient confirm your observations of the scatterplot from the previous section? How so, or why not? Discussion: Do the results of your linear regression and correlation analysis appear to confirm or contradict your initial belief that these variables were associated in some way? Critically evaluate this conclusion by addressing both the evidence in support of your conclusion about the purported relationship between your two variables, as well as cautions or problems with your data that would weaken your case. The form, shape, and strength of your scatterplot, as well as the strength of the correlation coefficient should be discussed in this evaluation. If your data contained outliers, these must also be discussed for full credit. Additionally, you must identify the sampling method you used (systematic, simple random, stratified, cluster, convenience, voluntary response, etc.) and discuss what limitations you see with your research, including sources of bias or other problems that might limit how well your research generalizes to the greater population. What further conclusions might you draw based on this deeper analysis? Technology Considerations: Discuss what technology you chose to use for your displays of quantitative data (Histogram and Boxplot) and why. Discuss what technology you chose to create your scatterplots and to compute the correlation coefficient and linear regression equation and why. For full credit be sure to discuss what technology options you considered, the pros and cons of each, and what considerations or concerns led you to make the choice you made. Project Part III: Statistical Testing (Inferential Statistics) points total) (38 In this section you will expand your project to explore inferential statistics for ONE of your quantitative variables: the one you decided in Part II was more nearly normal than the other. This section will include some external research, hypothesis testing and confidence intervals, and a discussion of hypothesis testing errors and issues. Your project should be submitted as a professional report including everything from Part I and II using the following headings: Research Proposal: all sections from Part I Data and Descriptive Statistics: all sections from Part II Part III: Statistical Testing -Research: Remind your reader which quantitative variable you considered most \"nearly normal\" in distribution and conduct some external research using the internet, library, or other academic sources to propose a reasonable guess for the true population average (mean) and standard deviation for your variable of interest. Be sure to remind your readers what population of interest you are referring to: for example, are you limiting your consideration to adult U.S. citizens, adults worldwide, U.S. college students, dogs owned as pets, etc. Describe the study/resource and why the proposed values are reasonable estimates for the population parameters (true mean and standard deviation). Cite your sources appropriately using APA style in-text citations or the Chicago Manual of Style footnotes and add a "References Sheet" on the last page. This section should be 1-2 paragraphs in length. **If you are able to obtain information about a plausible population mean but not a standard deviation, please use your sample standard deviation as an estimate. Sampling Distribution: Recall that the sampling distribution of means for a nearly normal variable should be Normal, with a mean equal to the population mean, and a standard deviation equal to the population standard deviation divided by the square root of your sample size. Naturally the true population parameters (mean and standard deviation) for your variable is unknown, however you identified a reasonable estimate in the section above. Based on these estimates, give the sampling distribution model that applies to your variable. Then use this sampling distribution to find the probability of seeing your data (sample mean) or more extreme by chance. Explain every step of your work and include a diagram of the sampling distribution normal curve and your observed sample statistic (use appropriate technology). One Sample Inferential Statistics: Suppose you suspect that the population mean found through your research is inaccurate and want to test the hypothesis that the population mean has changed (is different from the population mean found in your research). Since you are using quantitative data, you will be doing a 1 Mean Hypothesis Test (T-Test). You MUST do the following: 1-Mean Hypothesis Test Write your hypotheses using appropriate notation. Evaluate whether or not the conditions are met for you to conduct your test (you must actually verify all conditions - include diagrams or computations as applicable). o Use technology to conduct your test and professionally present your results. o o If your conditions are NOT met, you can proceed but you MUST discuss the implications of not having met conditions in your Discussion section. Indicate how you arrived at your results using that technology (i.e. give instructions, indicate the inputs, present screenshots if applicable). Present your results. Write a conclusion to your hypothesis test in the context of your research question using appropriate statistical terminology 1-Mean Confidence Interval Use technology to create a 95% confidence interval for the true population mean. o Show how you arrived at your results using that technology. o Respond to the question: Are the results of your confidence interval consistent with your hypothesis test? Why or why not? Discussion: Provide a detailed discussion of the potential issues and limitations of the results of your research into this one variable. Your discussion should answer all of the following questions (at a minimum!): How confident are you that your results are accurate and meaningful? Are your results statistically significant? How about practically significant (you'll need to refer to your confidence intervals to answer this question). What limitations should someone consider when looking at your research; for example: how well did the T-Test model apply to each test (how well were the conditions met) and how representative were your data. Discuss which potential error could have occurred in your research: Type I or Type II error and what it means in the context of your scenario. Suggest some possible reasons that this error might have occurred and what consequences might result from this error. What other problems do you see with your research that might limit how well your research generalizes to the greater population? Grading Rubric: Part III will be graded by components according to the following guidelines with comments provided to students. Research (4 points total) 4 Points 3 Points External resource is Proposed guess for clearly identified the true population and explained. mean is consistent with the external reference, but some details are missing. Proposed guess for the true population mean is justified and consistent with the Research findings external reference are based on used. somewhat legitimate sources and are referenced appropriately. Research findings are based on legitimate sources and are referenced Response is at times appropriately in-text difficult to and/or using understand, or at footnotes and a times uses 2 Points 1 Point Some evidence of external research is present. However, proposed guess for the true population mean is inconsistentwith the research findings or is not explained and/or justified. No evidence of No submi external research or submissio research is not plagiarize referenced at all. submissio match as Referencing skills need work. 0 Points reference page. statistical terminology inappropriately. Response is wellwritten, easy to understand, and uses correct statistical terminology. Sampling Distribution (SQR Criterion #2: 4 points total) Must identify the Sampling Distribution of the Mean and use this Sampling Distribution to find the probability of obtaining the sample mean (or more extreme) by chance. 4 Points 3 Points 2 Points 1 Point Sampling distribution is correctly identified. Sampling distribution is correctly identified. Uses incorrect strategies Cannot determine No submission, any strategy to submission is solve the problem. plagiarized, or submission doe not match assignment. OR Uses correct strategy to find the probability of the sample mean and expresses answers correctly. Diagram is included and is correct. Uses correct strategy but makes minor mistakes in finding or expressing answers (e.g. the probability of the sample mean). Uses correct strategy but problem solving process contains inaccuracies (such as sampling distribution not being correctly identified or other Diagram is missing major conceptual or or is inaccurate. procedural errors). 0 Points 1-Mean Hypothesis Test and 1-Mean Confidence Interval - SQR Criterion #1: Communicates mathematical and/or scientific concepts using appropriate symbols, notations and vocabulary. (4 points) 4 Points 3 Points 2 Points Hypotheses, checking assumptions, procedural steps, explanations, and conclusions use appropriate vocabulary, symbols, and notation with no errors. Hypotheses, checking assumptions, procedural steps, explanations, and conclusions Hypotheses, checking assumptions, procedural steps, explanations, and conclusions make poor or incorrect use of use appropriate vocabulary or vocabulary, contains many symbols, and errors. notation with no significant mistakes and/or with minimal inaccuracies. 1 Point 0 Points Hypotheses, checking assumptions, explanations, and conclusions are absent. No submission, submission is plagiarized, or submission does not match assignment. No use of statistical vocabulary and/or notation. 1-Mean Hypothesis Test and 1-Mean Confidence Interval - SQR Criterion #2: Applies appropriate process to solve the given problem. (4 points) 4 Points Hypothesis test and confidence interval 3 Points Hypothesis test and confidence interval computations computations use correct use correct strategy but strategy to makes minor find and mistakes in express finding or answers expressing correctly. answers. 2 Points 1 Point 0 Points Hypothesis test and confidence interval computations use incorrect strategies OR Uses correct strategy but problem solving process contains inaccuracies. Cannot determine any strategy to solve the problems. No submission, submission is plagiarized, or submission does not match assignment. 1-Mean Hypothesis Test and 1-Mean Confidence Interval - SQR Criterion #3: Analyzes, evaluates, justifies and interprets the reasonableness of a solution. (4 points) 4 Points 3 Points 2 Points 1 Point 0 Points Conclusions demonstrate accurate evaluation and interpretation of the results of the hypothesis test and confidence interval. Conclusions demonstrate accurate evaluation and interpretation of the results of the hypothesis test and confidence interval. Conclusions demonstrate insufficient or incorrect evaluation or interpretation of the results of the hypothesis test and/or confidence interval. Conclusions are not justified and/or results of the hypothesis test or confidence interval are not interpreted correctly. No submission, submission is plagiarized, or submission does not match assignment. Confidence interval results are used effectively to justify the reasonablenes s of the results of the hypothesis test and to form thoughtful and insightful conclusions. Confidence interval results are used to justify the reasonablenes s of the results of the hypothesis test. Confidence interval results are not correctly used to justify the reasonablenes s of the results of the hypothesis test. Some nuances in interpretation may be missed. Effective Use of Technology -- TC Criterion #2: Using, adapting or designing technologies to achieve the best results for research, communication or task-related objectives (4 points) 4 Points 3 Points 2 Points 1 Point Multiple, complementary technologies utilized to yield outstanding results, demonstrating advanced command of each of the tools. Multiple technologies used appropriately to complete project tasks in a manner sufficient to meet project goals, demonstrating proficiency in utilizing multiple tools. Project utilizes No technology some technology, used. but does not completely or consistently meet project goals. 0 Points No submissio submission i plagiarized, submission d not match assignment. Discussion (10 points) A complete response should address: statistical and practical significance, which type of error could have occurred, and a thorough exploration of limitations including the impact of sampling issues and how well the test/interval conditions were met. 10 Points 8 Points 6 Points 4 Points All questions are All questions are answered completely. answered. Most questions are Several questions answered. were left unanswered or responses contain major errors. Responses are wellResponses are Responses at written, easy to fairly well-written, times contain understand, and use and use mostly errors, are difficult correct statistical correct statistical to understand, Response terminology. terminology. and/or use demonstrates a incorrect statistical limited grasp of terminology. statistical concepts. Discussion of Discussion of significance, type and significance, type sources of error, and and sources of Discussion of research error, and significance, type research and sources of limitations is error, and thoughtful, correct limitations is research and/or reasonable mostly correct/ and demonstrates an reasonable but limitations are excellent grasp of may contain minor mostly correct/ statistical concepts. errors and reasonable and demonstrates a demonstrates a good grasp of fair grasp of statistical statistical concepts. concepts. Professionalism (3 points) 3 Points 2 Points Report is highly professional in appearance, typed using an appropriate mathematical typesetting program, easy to read/comprehend , and complete. Report demonstrates some professionalism, but contains distracting errors/problems in typesetting. 1 Point Report demonstrates some professionalism, but problems with organization, typesetting, and/or grammar Formatting/organizati make it difficult on, vocabulary, and/or grammar 0 Points Report does not demonstrate appropriate professionalism as required by the assignment. 2 Points Most question were left unanswered o responses are largely incorre do not make sense. Response demonstrates lack of understanding the relevant statistical concepts. need attention. Completeness (1 point) 1 Point 0 Points All required sections are present. Project is missing Parts I and II or other major sections. Parts I and II are included in the final project submission and are compatible (they all relate to one another). OR No submission / submission is plagiarized. to read