A population of 200 voters contains 52 Republicans, 128 Democrats, and 20 independents and members of other parties. A simple random sample of 20 voters will be drawn from this population. The expected number of Republicans in the sample is? and The standard error of the number of Republicans in the sample is?
To find the expected number of Republicans in the sample, we can use the concept of proportion.
First, we need to calculate the proportion of Republicans in the population:
Proportion of Republicans = Number of Republicans / Total population
= 52 / 200
= 0.26
Next, we multiply this proportion by the sample size to get the expected number of Republicans:
Expected number of Republicans in the sample = Proportion of Republicans * Sample size
= 0.26 * 20
= 5.2
Therefore, the expected number of Republicans in the sample is 5.2.
To calculate the standard error of the number of Republicans in the sample, we use the formula:
Standard error = sqrt((Proportion of Republicans * (1 - Proportion of Republicans)) / Sample size)
Using the values we have:
Standard error = sqrt((0.26 * (1 - 0.26)) / 20)
= sqrt(0.192 / 20)
= sqrt(0.0096)
= 0.098
Therefore, the standard error of the number of Republicans in the sample is approximately 0.098.
To find the expected number of Republicans in the sample, we can use the concept of probability. Since there are 52 Republicans in a population of 200 voters, the probability of selecting a Republican in a simple random sample is 52/200.
The expected number of Republicans in the sample can be calculated by multiplying the probability of selecting a Republican (52/200) by the sample size (20). Therefore, the expected number of Republicans in the sample is (52/200) * 20 = 5.2.
To find the standard error of the number of Republicans in the sample, we can use the formula:
Standard Error = sqrt((p * (1 - p) * (N - n)) / (N - 1))
Here:
p is the probability of selecting a Republican in the population (52/200)
N is the total population size (200)
n is the sample size (20)
Plug in these values into the formula, we get:
Standard Error = sqrt(((52/200) * (1 - 52/200) * (200 - 20)) / (200 - 1))
Simplifying this expression, we find:
Standard Error = sqrt((52 * 148 * 180) / (200 * 199))
Taking the square root of this expression, we get:
Standard Error ≈ sqrt(1441440 / 39800)
Standard Error ≈ sqrt(36.241)
Standard Error ≈ 6.02 (rounded to two decimal places)
Therefore, the standard error of the number of Republicans in the sample is approximately 6.02.