91Ó°ÊÓ

Parapsychology (psi) is a field of study that deals with clairvoyance or precognition. Psi made its way back into the news when a professional, refereed journal published an article by Cornell psychologist Daryl Bem, in which he claimed to demonstrate that psi is a real phenomenon. In the article Bem stated that certain individuals behave today as if they already know what is going to happen in the future. That is, individuals adjust current behavior in anticipation of events that are going to happen in the future. Here, we will present a simplified version of Bem's rescarch. (a) Suppose an individual claims to have the ability to predict the color (red or black) of a card from a standard 52 -card deck. Of course, simply by guessing we would expect the individual to get half the predictions correct, and half incorrect. What is the statement of no change or no effect in this type of experiment? What statement would we be looking to demonstrate? Based on this, what would be the null and alternative hypotheses? (b) Suppose you ask the individual to guess the correct color of a card 40 times, and the alleged savant (wise person) guesses the correct color 24 times. Would you consider this to be convincing evidence that that individual can guess the color of the card at better than a \(50 \sqrt{50}\) rate? To answer this question, we want to determine the likelihood of getting 24 or more colors correct even if the individual is simply guessing. To do this, we assume the individual is guessing so that the probability of a successful guess is \(0.5 .\) Explain how 40 coins flipped independently with heads representing a successful guess can be used to model the card-guessing experiment. (c) Now, use a random number generator, or applet such as the Coin-Flip applet in StatCrunch to flip 40 fair coins, 1000 different times. What proportion of time did you observe 24 or more heads due to chance alone? What does this tell you? Do you believe the individual has the ability to guess card color based on the results of the simulation, or could the results simply have occurred due to chance? (d) Explain why guessing card color (or flipping coins) 40 times and recording the number of correct guesses (or heads) is a binomial experiment. (e) Use the binomial probability function to find the probability of at least 24 correct guesses in 40 trials assuming the probability of success is 0.5 (f) Look at the graph of the outcomes of the simulation from part (c). Explain why the normal model might be used to estimate the probability of obtaining at least 24 correct guesses in 40 trials assuming the probability of success is \(0.5 .\) Use the model to estimate the \(P\) -value. (g) Based on the probabilities found in parts (c), (e), and (f), what might you conclude about the alleged savants ability to predict card color?

Step by step solution

01

Understanding the Hypotheses

For part (a), identify the null hypothesis (H_0) and alternative hypothesis (H_1). The null hypothesis (statement of no change) would be that the individual is simply guessing, which gives a success rate of 0.5. The alternative hypothesis is that the individual can predict better than by guessing.H_0: p = 0.5 (the individual is guessing)H_1: p > 0.5 (the individual can predict better)

02

Testing the Guesses

For part (b), the individual guesses correctly 24 times out of 40. To model this, consider 40 coin flips where heads represent a correct guess. Under the null hypothesis, the probability of a correct guess (heads) is 0.5. Calculate the probability of getting 24 or more heads if the individual is simply guessing.

03

Simulation with Coin Flips

For part (c), use a random number generator or an app like StatCrunch to flip 40 fair coins 1000 times. Record the proportion of times 24 or more heads appear. This gives an empirical probability of attaining 24 or more successes by chance.

04

Binomial Nature of the Experiment

For part (d), explain why this is a binomial experiment. The trial conditions satisfy binomial criteria: fixed number of trials (40), two outcomes (correct or incorrect guess), constant probability (0.5), and independence of trials.

05

Binomial Probability Calculation

For part (e), use the binomial probability formula to calculate the probability of getting at least 24 correct guesses out of 40 trials with a success probability of 0.5.

06

Normal Approximation

For part (f), explain why a normal model can be used. The binomial distribution approximates a normal distribution with large sample sizes. Compute the z-score for 24 correct guesses, and use the standard normal distribution to find the p-value.

07

Conclusion

For part (g), compare the results from steps (c), (e), and (f). Based on p-values and simulated proportions, conclude whether the observed 24 correct guesses provide convincing evidence of the individual's ability beyond random chance.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Null Hypothesis

In any experiment, the null hypothesis represents a statement of no effect or no change. In the context of this problem, the null hypothesis \(H_0\) asserts that the individual guessing the color of the card is purely random, with a probability of success \(p = 0.5\). This suggests that the individual has no special abilities and is merely guessing.

Alternative Hypothesis

Contrary to the null hypothesis, the alternative hypothesis posits that there is an effect or change. In this problem, the alternative hypothesis \(H_1\) claims the individual can predict the color of the card better than random guessing. Thus, the success rate would be higher than \(0.5\), indicating an ability beyond chance.

Probability Simulation

Probability simulation helps us understand the likelihood of certain outcomes under random conditions. In part (c) of the exercise, we simulate flipping 40 fair coins 1000 times to model the card-guessing experiment. Each flip representing a guess, the key here is to record how often we get 24 or more heads. This simulation helps us gauge if getting 24 successful guesses is plausible by random chance, providing a real-world probability to compare against our null hypothesis.

Binomial Distribution

A binomial distribution describes the number of successes in a fixed number of trials, each with the same probability of success. In our exercise, guessing the color of the card is akin to flipping a coin 40 times. Each trial has two outcomes (correct or incorrect guess), a constant probability of success (0.5), and trials are independent. Therefore, the number of correct guesses follows a binomial distribution with parameters \(n = 40\) and \(p = 0.5\). We can use this distribution to calculate the probability of getting at least 24 correct guesses.

Normal Approximation

When dealing with large sample sizes, the binomial distribution can be approximated by a normal distribution, which simplifies calculations. In our case, since \(n = 40\) and the probability of success \(p = 0.5\), the distribution of correct guesses can be approximated by a normal distribution with mean \( \mu = np = 20\textbackslash\) and standard deviation \( \sigma = \sqrt{np(1-p)} = \sqrt{10} = 3.16 \). To find the probability of at least 24 correct guesses, we compute the z-score and use the standard normal distribution tables to find the p-value. If this p-value is significantly low, it indicates the results are unlikely under the null hypothesis, suggesting the individual's guessing ability may be beyond chance.