stat act 6

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Feb 20, 2024

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BA 350: Statistics 1 Activity 6 (Chapter 4) 9/25/2020 Instructions: Complete the following questions. You may discuss with classmates, but you must turn in your own assignment. Unless otherwise specified, responses should be brief (one or a few sentences). Completed assignments should be submitted to Activity 6 on Schoology. Bayes’ Theorem and Probability Rules 1. Open the Excel file containing the class survey data (BA 350 Class Survey - Fall 2020 - Cleaned.xlsx), found in the Data and Excel Resources folder on Schoology. Create a PivotTable based on the data (Inset > PivotTable). Add Gender to the columns, Vacation to the rows, and Vacation (or Gender) to the Values of the PivotTable, to display a contingency table of these two variables. Use this contingency table to answer the following questions. a. What is the probability a BA350/PY203 student prefers to vacation to the beach, given that they are a male? 13/36= .36 b. What is the probability a randomly-selected BA350/PY203 student is a male who prefers to vacation to the beach? 13/71= .18 c. What is the probability a BA350/PY203 student is a male, given that they prefer to vacation to the beach? 36/13= 2.77 d. Is either gender independent of preferring a beach vacation? Show work to support your answer. Neither gender is independent of preferring a beach vacation. Male: 13/36= .36 Female: 14/ 35= .4

2. For four consecutive years, from 2014 to 2017, the Super Bowl coin toss at the start of the game landed tails each time. a. What are the chances of this occurring, assuming a “fair” coin (with 0.5 probability of landing heads) was used each year? There is a 50-50 chance that the coin used will land on either heads or tails. So, for 2014-2017 there were 100% tails 4 years/.5 probability= 8 b. Given that the coin toss landed tails for those four consecutive years, what was the probability that it landed tails again in the 2018 Super Bowl? 5 years/ .5 probability= 10 3. On Wednesday, we learned that the sensitivity of a COVID-19 diagnostic test is the probability of a positive result given that you have the virus. The specificity is the probability of a negative result given that you don’t have the virus. About a month ago, Abbott Laboratories announced the FDA approval of one of their tests, boasting a 97.1% sensitivity and a 98.5% specificity. We don’t definitively know what percent of the U.S. population has COVID-19, but suppose it is 5%, meaning that a randomly-selected person has a 5% chance of having the virus (in Bayesian statistical language, this is called your prior ). But if we know that you have tested positive using one of Abbott’s new tests, then what is the probability that you have the virus? a. First, define your events using capital letters (e.g. A and B). A= have covid A’= do not have covid B= positive test B’= negative test

b. Now, find the percent of people who actually have the virus among those who test positive using one of Abbott’s new tests. Show your work using Bayes’ Theorem. Hint: The complement rule works for conditional probabilities as well. So, for instance, , where is the complement of P(A|B)=? P(B|A)= .971 P(A)= .05 P(A’)= .95 P(B’|A’)= .985 P(B|A’)= 1-.985= .015 = 77.3% (.971)(.05) ((.971)(.05))+((.015)(.95)) 4. Bonus (optional): From 2014 through 2020, the Super Bowl coin toss landed heads just once. Suppose a friend of yours plans to bet money on the 2021 Super Bowl coin toss (assuming that there is one!) landing heads, reasoning that it is due to land heads because of the string of so many tails in recent years. Statistically speaking, how would you advise your friend? 7 years 6/7 tails = .86 = 86% I would advise them to bet on the coin landing on tails because in the past 7 years, there has been a .86 probability that the coin landed on tails. This is an 86% chance of landing on tails, which is a very high probability.

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