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 _____ ?
Q1. A population has a mean of µ = 80 with σ = 20.
a. If a single score is randomly selected from this population, how much distance, on average, should you find between the score and the population mean?
b. If a sample of n = 4 scores is randomly selected from this population, how much distance, on average, should you find between the sample mean and the population mean?
c. What is the probability that sample mean will be less than 70 for a sample of 16 scores?
To find the expected number of Republicans in the sample, we can use the concept of probability.
Step 1: Calculate the probability of selecting a Republican in a single draw from the population.
- There are a total of 200 voters in the population.
- There are 52 Republicans in the population.
- Therefore, the probability of selecting a Republican in a single draw is 52/200 or 0.26.
Step 2: Multiply the probability from Step 1 by the sample size to find the expected number of Republicans in the sample.
- The sample size is 20 voters.
- Therefore, the expected number of Republicans in the sample is 0.26 * 20 = 5.2.
So, the expected number of Republicans in the sample is 5.2.
To find the standard error of the number of Republicans in the sample, we need to use the formula for the standard error of a proportion.
The formula for the standard error of a proportion is given by:
- SE = sqrt(p * (1-p) / n)
Where:
- SE is the standard error
- p is the probability of success
- n is the sample size
In this case, we have already calculated the probability of selecting a Republican in a single draw as 0.26, and the sample size is 20.
Let's plug in these values into the formula to find the standard error:
- SE = sqrt(0.26 * (1-0.26) / 20)
- SE = sqrt(0.26 * 0.74 / 20)
- SE = sqrt(0.1924 / 20)
- SE = sqrt(0.00962)
- SE ≈ 0.0981
So, the standard error of the number of Republicans in the sample is approximately 0.0981.