During a weekend music festival, surveys are taken from a sample size of 40 people on their age. The sample proportion of people in their 40s compared to everyone is found. What is the distribution of the sample proportion?

Question

(1 point)
• distribution skewed to the left
• distribution skewed to the right
• uniform distribution
• normal distribution

Answer 1

To determine the distribution of the sample proportion in this scenario, we need to consider the characteristics of sample proportions in general. The distribution of the sample proportion (often denoted as \( \hat{p} \)) can be approximated using the normal distribution when certain conditions are met, particularly when the sample size is sufficiently large and when the expected number of successes (in this case, people in their 40s) and failures are both greater than 5.

In this case, if we have a sample size of 40 people and we are looking at the proportion of people in their 40s, the distribution of the sample proportion will tend to be normal if:

1. The sample size is large enough.
2. The number of people in their 40s (\( np \)) and not in their 40s (\( n(1-p) \)) are both greater than 5.

Since the sample size is 40, if the proportion of people in their 40s is not too extreme (e.g., around 0.5), the distribution of the sample proportion can be approximated by a normal distribution.

Therefore, the most accurate answer, assuming these conditions are satisfied, is:

• normal distribution