Among employed women, 25% have never been married. Select 10 employed women at random. What is the mean number of women in such a sample who have never been married? What is the standard deviation?
I think the mean would be 2.5, but I'm not sure about the standard deviation.
probability that the women was unmarried = p = 0.25
probability that she was married = q = 1-p = 0.75
sample size , n = 10
mean = np = 10 x 0.25 = 2.5
standard deviation = sq. root (npq)
=1.37
To find the mean and standard deviation for the number of employed women who have never been married in a random sample of 10 women, we can use the binomial distribution formula.
Let's start by calculating the mean (expected value):
Mean (μ) = n * p
Where:
n = number of trials (10 women)
p = probability of success (25% or 0.25)
So, μ = 10 * 0.25 = 2.5
The mean number of women in the sample who have never been married is indeed 2.5.
Next, let's calculate the standard deviation (σ):
Standard Deviation (σ) = √(n * p * (1 - p))
σ = √(10 * 0.25 * (1 - 0.25)) = √(2.5 * 0.75) ≈ √1.875 ≈ 1.37
Therefore, the standard deviation for the number of women in the sample who have never been married is approximately 1.37.
To calculate the mean number of women in the sample who have never been married, we need to multiply the percentage (25%) by the sample size (10). This will give us the expected number of women in the sample who have never been married.
Mean = Percentage * Sample Size
Mean = 25% * 10
Mean = 0.25 * 10
Mean = 2.5
You are correct that the mean number of women in the sample who have never been married is 2.5.
Now let's find the standard deviation. Since we have a binary outcome (either the women are married or not married), we can model the distribution as a binomial distribution. The standard deviation for a binomial distribution can be calculated using the following formula:
Standard Deviation = sqrt(n * p * (1 - p))
Where:
- n is the sample size (10 in this case)
- p is the probability of success (25% or 0.25 in this case)
Standard Deviation = sqrt(10 * 0.25 * (1 - 0.25))
Standard Deviation = sqrt(0.25 * 0.75)
Standard Deviation = sqrt(0.1875)
Standard Deviation ≈ 0.433
So, the standard deviation for the number of women in the sample who have never been married is approximately 0.433.