Simulation: Predicting the Future 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 research.
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.

Verified step by step guidance

1. Step 1: Recognize that the problem involves a binomial distribution because each trial has two possible outcomes (correct guess or incorrect guess), with a fixed number of trials (n = 40) and a constant probability of success (p = 0.5).
2. Step 2: Understand that the normal model can be used to approximate the binomial distribution when the sample size is large enough. A common rule of thumb is that both np and n(1-p) should be at least 10. Calculate np = 40 \times 0.5 and n(1-p) = 40 \times 0.5 to verify this condition.
3. Step 3: Since the condition is met, approximate the binomial distribution with a normal distribution having mean \(\mu = np\) and standard deviation \(\sigma = \sqrt{np(1-p)}\).
4. Step 4: Apply the continuity correction by adjusting the value of interest (24 correct guesses) by 0.5 to better approximate the discrete binomial with the continuous normal distribution. For "at least 24", use 23.5 as the cutoff point.
5. Step 5: Calculate the z-score using the formula \(z = \frac{23.5 - \mu}{\sigma}\), then use the standard normal distribution table or a calculator to find the probability corresponding to this z-score. This probability is the estimated P-value for obtaining at least 24 correct guesses.

Key Concepts

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. In this question, guessing correctly in 40 trials with a 0.5 success probability fits a binomial setting. Understanding this distribution helps calculate exact probabilities for outcomes like getting at least 24 correct guesses.

Normal Approximation to the Binomial

When the number of trials is large, the binomial distribution can be approximated by a normal distribution with mean np and variance np(1-p). This simplifies calculations, especially for cumulative probabilities. Here, with 40 trials and p=0.5, the normal model is appropriate to estimate the probability of 24 or more successes.

P-value and Hypothesis Testing

A P-value measures the probability of observing data as extreme as, or more extreme than, the actual results under the null hypothesis. In this context, it quantifies how likely it is to get at least 24 correct guesses by chance if the true success rate is 0.5. A small P-value suggests evidence against the null hypothesis.