## Distributions of Data and the Normal Distribution

Distributions of Data and the Normal Distribution

###### Purpose: In this assignment you will explore how to use fundamental analytical tools to understand how distributions of data work in statistics. This includes how to compute basic probability and how the normal distribution lets us compare two sets of data that may not use the same scale.

The mastery standards covered by this assignment are

S6 –   Distributions of Data: Students will be able to describe distributions of data.

S7 –   Normal Distributions of Data: Students will be able to use the properties of normal distributions of data and explain the results of that use in context.

MP2 – Communicate a Viable Argument – Entire assignment: Students will be able to justify a statistical analysis in words and communicate that justification.

# Instructions:

Note that this is an INDIVIDUAL assignment, and you are responsible to do your own write-up. You may have started to work on this TAMS in class as a pair or group, but you are to work on this on your own at home. Although you are allowed to discuss this assignment with other class members, you may NOT work together on an electronic document. “Working together” in this context is ONLY by talking, not by typing. You MUST do your own work in the software, save your own copies of the graphics, and produce your OWN written document. Documents that begin as a shared document and are modified will NOT be accepted and will result in a mastery grade of 0 for everybody involved.

You are to submit your answers using the template posted with the assignment. Your template needs to be submitted online through Canvas.

Problem 1

As of March 22, 2020, Slovenia reported a total of 414 cases of COVID-19, of which 412 have been classified by both gender and age range. The data is summarized in the table below (Source: https://www.statista.com/statistics/1104413/slovenia-confirmed-coronavirus-cases-covid-19-by-age-and-gender/).

Use this table to answer the questions below. Before doing so, fill in the missing totals in this table, then answer the questions below.

1. What is the percentage of female COVID-19 cases?
2. What is the percentage of COVID-19 cases in the age range from 35 to 74?
3. What percentage of female COVID-19 cases are younger than 60 years?
4. What percentage of the 15 to 34-year-old COVID-19 cases are men?
5. According to this data, which age group has the lowest risk? Which gender has the highest risk? Is there a particular age and gender combination that is the riskiest group? Give reasons for your answers and show any computations you may have performed.

Problem 2

One of the key purposes of statistical analysis is to make inferences. According to the dictionary, inference is a conclusion reached on the basis of evidence and reasoning. Specifically, for statistics, what we want to do is to be able to choose a sample, obtain some data about that sample, and then come to a conclusion about the bigger population, based on just the information from the sample.

How can it be possible to take a small sample (say 100 people) and conclude something about millions of people? The answer lies in a combination of probability and the normal distribution. In this problem, we are going to explore the normal distribution in a simplified fashion. The normal distribution rarely shows up in real life. But as you will discover when we learn about sampling distributions (coming up), the normal distribution shows up a LOT when we look at groups of subjects in a statistical study.

One place that the normal distribution does show up in the “real world” is in standardized testing, particular the SAT test.

SAT Data

Based on the data from the most recent test scores of 2019 high school graduates, the mean score and standard deviation for the overall test (composite) score and the scores for the English and Math portions are shown below.

You have been hired as an admissions evaluator for a large national university. Their guidelines state that students who scored in the top 10.2% of test takers based on their composite score, should be automatically passed to the next step for being considered for admission. Additionally, any student scoring in the top 4.09% in Math also gets automatically passed to the next step. Because you have taken statistics, your supervisor has asked you to write up a set of guidelines that other evaluators will use to quickly look at hundreds of applications to determine if they automatically get moved along in the process.

1. Create a table of cut-off scores for evaluators to use that would allow them to quickly determine if an application qualifies for automatically moving along. In particular, a table with the various cutoff scores for each type of section or composite score would be very helpful. Remember, in order to convert percentages to scores you will need to use a z-score table to get z-scores and then use the z-score formula to find the necessary cutoff scores. Show all your work and computations. Explain how the scores you calculated meet the guidelines.

2. Write a short memo to your supervisor which has a short introductory paragraph, then shows your table of cut-off scores.
• Your supervisor was told by her boss that not enough students are automatically passed on to the next step and wants your supervisor to come up with new cut-off scores. However, your supervisor just wants to change ONE cut-off score so that the admission officers who have used your table do not have to learn a whole new set of cutoff scores. The choices are to lower the math SAT cut-off score by 50 or composite score by 75.

Your supervisor asks you which change would give “the biggest bang for the buck”. Specifically, your supervisor asks you to compute the percentage of students who would automatically be passed to the next stage under these new cut-off scores, and then to identify which of the two cut-off changes increases the percentage of students passed on to the next step by the most.

Fill in the table below and show (below it) how you computed one of the percentages, as well as how you computed the increased percentage of students who are being passed along automatically. State your conclusion as to which cut-off score should be changed and why.

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