Final answer:
To answer the student's question on probability, one must calculate the standard error of the sample proportion, find the z-scores for the population proportion plus or minus 3%, and then look up these probabilities in the standard normal distribution. The answer to the probability question will be expressed to four decimal places.
Step-by-step explanation:
The student's question involves using the Central Limit Theorem to calculate the probability that the sample proportion of Grammy award-winning American singers differs from the population proportion by less than 3%. We are given that 40% of American singers are Grammy award winners, and we need to find the probability that in a random sample of 743 singers, between 37% (0.40 - 0.03) and 43% (0.40 + 0.03) are Grammy winners.
Since the sample size is large enough, we can assume that the sampling distribution of the proportion is approximately normal. We can use the formula for the standard error of the proportion, given by SE = √(p(1-p)/n), where p is the population proportion and n is the sample size, to calculate the standard error. Then, we can find the z-scores corresponding to the population proportion plus or minus 3% and use the standard normal distribution to find the probability of being within these z-scores.
The process involves the following steps:
- Calculate the standard error (SE).
- Determine the z-scores for p - 0.03 and p + 0.03.
- Use the z-scores to find the probability using the standard normal distribution.
After conducting the above calculations, the required probability can be found and should be rounded to four decimal places as per the student's request.