An ESP study discovers a subject whose performance appears to be better than guessing. The experimenter looks at many cards bearing one of five shapes (star, square, circle, wave, and cross) in an order determined by random numbers. The subject cannot see the experimenter to avoid any nonverbal clues. The answers of a non-ESP subject should be independent observations, each with probability 1/5 of success. We record 1000 samples. What are the mean and standard deviation of the proportion of successes among the 1000 attempts?
The mean should be (1/5)*1000 = 200 correct guesses and the standard deviation should be
sqrt [p(1-p)N] = sqrt[(1/5)(4/5)1000] = 12.65
To find the mean and standard deviation of the proportion of successes among the 1000 attempts, we can apply the concepts of probability and statistics. Here's how we can approach it step by step:
Step 1: Understanding the problem
In this experiment, the subject is shown cards with five different shapes (star, square, circle, wave, and cross) in a random order. The subject's task is to identify the shape on each card, without any external clues. The success is defined as correctly identifying the shape.
Step 2: Calculating the probability of success
Since the subject is choosing from five options (shapes) randomly, the probability of any single guess being correct is 1/5 or 0.2.
Step 3: Generating 1000 samples
We have conducted 1000 trials, so we can consider each attempt as a sample. In these samples, we record the number of successes (correct guesses).
Step 4: Calculating the proportion of successes
For each of the 1000 samples, divide the number of successes by the total number of attempts (which is 1000 in this case). This will give us the proportion of successes in each sample.
Step 5: Calculating the mean
To find the mean of the proportion of successes, sum up all the proportions calculated in step 4 and divide by the total number of samples (which is 1000).
Step 6: Calculating the standard deviation
To find the standard deviation of the proportion of successes, we need to calculate the variance first. For each of the proportions calculated in step 4, subtract the mean (calculated in step 5) and square the result. Sum up all these squared differences and divide by the total number of samples (1000). Finally, take the square root of this variance to get the standard deviation.
So, by following these steps, you can obtain the mean and standard deviation of the proportion of successes among the 1000 attempts in your experiment.