Standards Addressed
Since statistics is an elective course, there are no state standards that apply to the content taught. However, the school compiled a list of essential learner outcomes that I addressed:
Central Limit Theorem
Control charts
Binomial Distributions
Confidence intervals (z and t intervals with 1 or 2 variables w/quantitative and categorical data)
Test of significance aka Hypothesis testing (z and t intervals with 1 or 2 variables w/quantitative and categorical data)
Two way tables and Chi-square test
There are also similar standards presented by the AP Statistics rubric (though this is not an AP class). We addressed many components of the following (note that initial text by AP was reworded as goals): 1E: Students will be able to visually summarize categorical data using bar charts, pie charts, frequency tables, and two-way tables.
2C: Students will be able to plan and conduct an experiment to identify causation.
2D: Students will be able to determine how their study, survey, or experiment results generalize to a population.
3D: Students will be familiar with common sampling distributions and how the Central Limit Theorem changes them.
4A: Students will be able to estimate means and proportions, and differences of two means or proportions, at a point and with a margin of error in a confidence interval.
4B: Students will be able to state hypotheses and perform a test of significance on means, proportions, and differences of two means or proportions.
Instructional Format and Objectives Layout
During the instructional part of the unit (the first 3 weeks), I lectured for half of the period on one day, allowed time in class to work on homework the next day, and we took the quiz on the third day. After each quiz, I began lecturing on the new topic. This was repeated for the five chapters. The daily objectives for the lecture days for each of the chapters are below. Two-way tables
Read and construct a two-way table of counts.
Find a conditional distribution from a two-way table.
Find a marginal distribution from a two-way table.
Identify Simpson’s Paradox from a pair of two-way tables.
Binomial distribution
Understand the conditions for using the binomial distribution.
Calculate the probability of a single value occurring in a binomial distribution (using binompdf on TI-83).
Calculate the probability of a range of values occurring in a binomial distribution (using binomcdf or 1 – binomcdf on TI-83).
Confidence intervals
Students will be introduced the idea of uncertainty in sample data.
Students will understand that increasing the sample size or decreasing the confidence level will reduce the amount of uncertainty in sample data.
Students will see the main formula for calculating a confidence interval and know how to use it.
Hypothesis testing
Students will understand why it is necessary to state a level of significance with a claim.
Students will be able to state a scenario (test) and its null hypothesis.
Students will understand the difference between the null and alternative hypotheses and how each is used.
Students will understand when to use a one-sided or two-sided test of significance.
Students will be introduced to the calculation of a p-value for a one-sided and two-sided test of significance.
Power and errors
Students will understand the difference between Type I and Type II errors.
Students will associate the probability of a Type I error to the familiar variable α.
Students will associate the probability of a Type II error to the new variable β.
Students will be introduced to the idea of power.
Students will be able to calculate the probability of a Type II error given power.
In the second half of the unit, the first days were used to prepare for game day with students working in groups. The two game days involved students pairing up with another group, playing the game, recording their data, and switching to the next group. The writing days were spent in the computer lab working in groups at the computers on Google Docs. During all days, I briefly introduced new activities up front before floating between groups as questions arose.
Objectives were made clear to students using the rubric: each team knew exactly what they needed to accomplish. The objectives were based on the five chapters recently covered in class, so even though their understanding was not yet sound, students were familiar with the topics. The assessed items listed below are based on the rubric that all students received at the start of the project.
Game play
Do your directions clearly describe the task your subjects are asked to do?
Is your spreadsheet setup to capture every subject on one horizontal row?
Do you have (a) column(s) to capture every piece of quantitative data you will need?
Do you have (a) column(s) to capture every piece of qualitative or success-fail data you will need?
Did you correctly obtain the mean and standard deviation of the population?
Did you select an SRS of 8 participants and copy them to a new tab of the spreadsheet?
Do you have all the materials required to play your game (with backups if things are breakable)?
Did you participate in all of the other student’s games?
Did you record the raw data for all participants during the periods?
Introduction
Concisely describes the study, purpose, and results in a few sentences.
Includes a short introduction that describes what motivated your specific study.
Clearly explains what you did to the point where the study could be replicated by a reader.
Two-way table
Present in paper with at least 2 rows and 2 columns, expressing counts of categorical data.
Marginal totals of the counts are present and correct.
Binomial distribution
A correctly generated graph of a binomial distribution is present in the paper.
Graph is properly labeled.
Binomial distribution is based on a fixed number of trials, fixed probability, and measures outcomes with success or failure.
Confidence interval
Interval is present in paper with calculation displayed and described.
Correctly uses the sample mean as the center and z* value for the given level of confidence.
Discusses validity of the confidence interval – is it based on a true SRS? Is there non-response or undercoverage bias? What about the effects due to randomness?
Hypothesis testing
An alternative hypothesis (and its corresponding null) is posed as a prediction before the results are found.
An alpha level is chosen before results are found.
Z-score and p-value are calculated and all steps are shown.
You reject or fail to reject the null hypothesis based on the comparison of your p-value and alpha level.
Error
Problems from the study, such as leaking bottles, moved containers, crumbling cookies, or clever rule-benders are thoroughly discussed and addressed as excusable error or something to seriously consider when interpreting results.
Conclusion
Includes a short conclusion that discusses the limitations of the study and points to future studies that could be done to further this research.
Writing style
Sections are clearly divided with descriptive headings (including an abstract, introduction, procedural section, results section, and conclusion).
Written in first-person active voice (I did, we did) and NO PASSIVE VOICE.
Ideas logically flow throughout the paper.
No "fluff" or grandiose introductions to sections or wasted words.
Writing flows between sections (doesn’t feel like 3-4 people wrote it).
Population
Evaluation of sample data vs. population: Did your actual population mu fall within your predicted confidence interval?
Evaluation of sample data vs. population: Given the alpha value you chose in your sample paper, did you discuss the probability of a Type I error? If you commit a Type II error (failing to reject the null when it actually varied in the population), did you mention it and explain? If you correctly accepted or rejected the null, did you mention this and explain?
Histogram: Did you correctly plot each data point of a quantitative variable in a histogram/pdf plot?
Histogram: Did you compute the sample mean, x-bar, of 24-28 different SRS clumps of 8 students each and correctly plot them in the same histogram/pdf plot?
Team contribution
You created quality work for the team project.
You completed work on time for the needs and deadlines of the group.
You took a leadership/captain role on an even number of tasks.
You shared responsibility of the project (did not hog all of the work).
You contributed an equal amount of effort to the team’s success.