55% of the people in a town have brown eyes. you choose a random sample of 200 people to survey regarding their eye color. What is the mean and standard deviation of the proportion of people in your sample that have brown eyes?
To find the mean and standard deviation of the proportion of people in your sample that have brown eyes, we will use the formulas for calculating the mean and standard deviation of a proportion.
Mean (Expected Value):
The mean of a proportion is simply the proportion itself. In this case, the proportion of the people in the town that have brown eyes is 55%. Therefore, the mean of the proportion of people in your sample with brown eyes is also 55%.
Standard Deviation:
The standard deviation of a proportion can be calculated using the formula:
σ = √((pq)/n)
Where:
- σ is the standard deviation of the proportion
- p is the proportion of success (55% in this case, which is 0.55)
- q is the proportion of failure (100% - 55% = 45%, which is 0.45)
- n is the sample size (200 people in this case)
Now, let's calculate the standard deviation:
σ = √((0.55 * 0.45) / 200)
σ = √(0.2475 / 200)
σ ≈ √0.0012375
σ ≈ 0.0351705
Therefore, the mean of the proportion of people in your sample that have brown eyes is 55%, and the standard deviation of that proportion is approximately 0.03517.