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 ______ ?
The standard error of the number of Republicans in the sample is _____ ?
To find the expected number of Republicans in the sample, you can use the concept of proportion. In this case, you need to calculate the proportion of Republicans in the population and then multiply it by the sample size.
Step 1: Calculate the proportion of Republicans in the population.
The proportion of Republicans in the population can be found by dividing the number of Republicans by the total population size.
Proportion of Republicans = Number of Republicans / Total population size
Proportion of Republicans = 52 / 200 = 0.26
Step 2: Multiply the proportion of Republicans by the sample size.
Expected number of Republicans in the sample = Proportion of Republicans * Sample size
Expected number of Republicans in the sample = 0.26 * 20 = 5.2
Therefore, the expected number of Republicans in the sample is 5.2.
Now, let's calculate the standard error of the number of Republicans in the sample. The standard error measures the amount of variability or uncertainty in the estimate of the expected number of Republicans.
Standard error is calculated using the formula:
Standard Error = sqrt[(Proportion of Republicans * (1 - Proportion of Republicans)) / Sample size]
Using the given values:
Proportion of Republicans = 0.26
Sample size = 20
Standard Error = sqrt[(0.26 * (1 - 0.26)) / 20]
Standard Error = sqrt[(0.26 * 0.74) / 20]
Standard Error = sqrt[0.1924 / 20]
Standard Error = sqrt(0.00962)
Standard Error = 0.098
Therefore, the standard error of the number of Republicans in the sample is approximately 0.098.