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Preparing for the next class

In the next in-class activity, you will need to identify observational units and types of variables, describe how to simulate a random process using a coin, and determine whether data provide evidence against a null hypothesis based on a P-value.

Questions 1–6: After taking several exams in a math course, a student suspects that the majority (more than half) of true/false questions on exams have a correct answer of “false.” To investigate this hypothesis, she takes a sample of 16 true/false questions from previous exams and determines whether the correct answer for each question is “false.”

Question 1

1) What are the observational units in this study?

a) Exams

b) Questions

c) True or false

d) Students

Question 2

2) What is the sample size?

Question 3

3) What variable is measured on each observational unit? Is the variable categorical or quantitative?

Question 4

4) If the two answer options, “true” or “false,” were equally likely to be the correct answer, how many of the 16 questions would you expect to have “false” as the correct answer? Explain.

Question 5

5) If the two answer options, “true” or “false,” were equally likely to be the correct answer, describe how you could use a coin to simulate sampling 16 questions and countingthe number of those questions with “false” as the correct answer.Clearly explain what each coinfliprepresents, what a “heads” or “tails” outcome would represent, and how you would calculate the sample result.

Question 6

6) Go to the DCMP Sampling Distribution of the Sample Proportiontoolat https://dcmathpathways.shinyapps.io/SampDist_prop/.You will use this toolto simulate the coin-tossing process you described in Question 5.Enter the following inputs:

•Set the Population Proportionto0.5. This represents the probability that a coin lands on “heads.”•Set the Sample Sizeto16. This is the number of times we would like to flip the coin.•Under “Select how many samples you want to simulate drawing from the population,” select “1.”

Part A: Click “Draw Sample(s).” This will generate one sequence of 16coin tossesand plot a bar graph of the results (the Data Distribution), where “1” represents “heads” or a “success” and “0” represents “tails” or a “failure.”If “heads” represents that the correct answer to a true/false question is “false,” how many questions in this simulation had a “false” correct answer?

Hint: The tool displays the number of “successes” and the number of “failures” in the caption of the Data Distribution bar graph.

Part B: Click “Reset,” select “1,000”under“Select how many samples you want to simulate drawing from the population,” and click “Draw Sample(s).” This will repeat the simulation of sampling 16 true/false questions 1,000 times. Select “Show Distribution of Successes.” The “Sampling Distribution of Number of Successes” shown at the bottom of the page displays how the number of true/false questions with “false” as the correct answer varies across these 1,000 trials of selecting 16 questions. Where is this distribution centered? Does this make sense?Explain.

Part C: In the student’s sample of 16 questions, she found that “false” was the correct answer in 11 of the questions. Use the “Find Probability for Samp. Dist.” option in the tool to count the proportion of simulated samples that had 11 or more questions with “false” as the correct answer.

Hint: The tool will only calculate the proportion of samples that are at or below a value. Usethetoolto find the proportion of samples that had 10 or fewer questions with “false” as the correct answerand then subtract this proportion from 1to obtain your answer.

Part D: Based on your answer in Part C, do you think the student’s data provide evidence that more than half of the true/false exam questions in this course have “false” as the correct answer? Explain how you are using the answer in Part C to determine your answer.