5. To investigate the idea that Joy was just guessing which shirt was worn by which type of person, we will begin by assuming that the null hypothesis is true. The exercise below represents the outcome you would expect if Joy was guessing by chance. e Get a couple sheets of paper and cut them into 24 \"cards.\" On half of the cards write \"Parkinson's\" on the other half write \"None.\" e Turn the cards face down and mix them up. Randomly pick out 12 cards. Before turning each card over, \"smell\" and guess whether each of the 12 cards will say \"Parkinson's\" or \"None\". Record the number of correct and incorrect guesses you made. Reshuffle the cards and repeat a couple of more times. Trial # Tally of correct Tally of incorrect identifications identifications 6. Create a dotplot of the number of correct identifications in our class: 7. In the actual experiment, Joy identified 11 of the 12 shirts correctly. Based on the very small-scale simulation you did, what proportion of the simulations resulted in 11 or more shirts correctly identified, assuming that the person was guessing? 8. The proportion you just calculated is a crude estimate of a true probability called a p-value (short for probability-value). How might we improve our estimate of the true probability? TPS5 Doug Tyson, Central York HS tyson.doug@gmail.com STATISTICAL INFERENCE FROM THE SIMULATION . Use this web site to run a simulation of this activity 10,000 times: https://www.lock5stat.com/StatKey/ Choose "Confidence Interval single Proportion." Then choose "Edit Data" and enter 5 AND 12 for the count and sample size. Then run this simulation 10,000 times. Change the dotplot display from "Proportion" to "count." This shows what the distribution looks like IF Joy was simply guessing. This should be a better estimate of the p-value for 11 or more shirts correctly identified, assuming that this person was just guessing. Bootstrap Dotplot of Count - 3000 OLeft Tail Two-Tail Right Tail samples - 12060 mean = 5.979 500 stad error - 1.731 2000 1500 1000 500 11 9. By just looking at your graph, answer the following questions . Is it possible that Joy correctly identified 11 out of 12 shirts just by random chance (guessing)? Is it likely? 10. Now we want to figure out a more accurate p-value. So first step - Joy guessed 11/12, so do we want to look for the probability of obtaining this score in the right-tail, the left-tail, or both tails (two-tails)? Hint: Would Joy's score be found above or below the mean, or in either tail? 11. This is a one-tailed test, specifically a right-tailed test. Now click the check-box that says "right-tail" - this is where we will look for the probability of guessing 11/12. When you click this box, it shows you the probability of obtaining a given result. . In the bottom text box that pops up, click on it and type in 11 . What proportion is shown that corresponds to correctly identifying 11/12 Parkinson's patients? write your answer p = ?SMELLING PARKINSON'S DISEASE INTRODUCTION As reported by the Washington Post, Joy Milne of Perth, UK, smelled a \"subtle musky odor\" on her husband Les that she had never smelled before. At first, Joy thought maybe it was just from the sweat after long hours of work. But when Les was diagnosed with Parkinson's 6 years later, Joy suspected the odor might be a result of the disease. Scientists were intrigued by Joy's claim and designed an experiment to test her ability to \"smell Parkinson's.\" Joy was presented with 12 different shirts, each worn by a different person, some of whom had Parkinson's and some of whom did not. The shirts were given to Joy in a random order and she had to decide whether each shirt was worn by a Parkinson's patient or not. e How many correct decisions (out of 12) would you expect Joy make if she couldn't really smell Parkinson's and was just guessing? e How many correct decisions (out of 12) would it take to convince you that Joy really could smell Parkinson's? SIMULATING THE EXPERIMENT Although the researchers wanted to believe Joy, there was a chance that she may not really be able to tell Parkinson's by smell. It's logical to be skeptical of claims that are very different than our experiences. If Joy couldn't really distinguish Parkinson's by smell, then she would just have been guessing which shirt was which. The researchers were not willing to commit time and resources to a larger investigation unless they could be convinced that Joy wasn't just guessing. When researchers have a claim that they suspect (or hope) to find evidence against, it's called the null hypothesis. e What claim were the researchers hoping to find evidence against? That is, what was their prior belief (null hypothesis) about the ability to smell Parkinson's? e What claim were the researchers hoping to find evidence for? This is called the alternative hypothesis or the research hypothesis. e We will now test this hypothesis determining the probability of Joy correctly identifying 11 out of 12 patients by chance alone. We are going to set up an expectation right now. We want to be very sure we are not making a false claim about Joy's abilities, so in order to claim that Joy is not guessing (and reject the null hypothesis) she has to identify so many patients_that there is only a 1% chance that she did it by chance alone. Our alpha level is p = 0.01 TPS5 Doug Tyson, Central York HS tyson.doug@gmail.com 12. An interesting side note is that Joy's one \"mistake\" really wasn't a mistake. The shirt was worn by a person who supposedly didn't have Parkinson's even though Joy claimed that she could smell the telltale smell on that shirt. That person called the experimenters 8 months after the experiment and reported that he had just been diagnosed with Parkinson's disease. That meant that Joy correctly identified 12 out of 12 shirts. What is the approximate p-value for 12 shirts correctly identified, assuming that this person was just guessing? NOTE: A small p-value is considered strong evidence against the null hypothesis and in favor of the alternative hypothesis. But how small is small? As a rule of thumb, statisticians generally agree that p-values below 0.05 provide pretty strong evidence against the null hypothesis. Observed results with small p-values are said to be statistically significant. Making a decision 13. So going back to our null hypothesis - based on the probability you found, how likely is it that Joy correctly guessed 12/12 Parkinson's patients by chance? We will use this probability to either reject the null hypothesis or keep the null hypothesis. Your choices: a) There is a very small likelihood that Joy could have correctly identified 12 of 12 patients by chance - less than a 1% chance. Therefore we reject the null hypothesis. b) The chance that Joy was able to correctly identify 12 of 12 Parkinson's patients by guessing alone is greater than a 1% chance. Therefore we retain the null hypothesis. TPS5 Doug Tyson, Central York HS tyson.doug@gmail.com