Final answer:
The variance for the number of favorable voters in groups of 21, where 22% of the population is in favor, can be calculated using the binomial distribution variance formula. The variance is 3.7992, which rounds to 3.8, and the closest given option is 3.6.
Step-by-step explanation:
The question asks for the variance for the number of voters who favor a measure where 22% of the voters are in favor. To find the variance for the number of favorable voters in groups of 21, we can use the binomial distribution variance formula, which is variance = n × p × (1 - p) where n is the number of trials (voters in the group), and p is the probability of success (a voter favoring the measure).
In this case, with n = 21 and p = 0.22, the calculation for variance would be 21 × 0.22 × (1 - 0.22). Carrying out this calculation:
Variance = 21 × 0.22 × 0.78 = 3.7992, which can be rounded to 3.8. Therefore, the correct answer is not explicitly stated among the options given (A: 3.6, B: 4.6, C: 1.9, D: 13), but the closest option is A) 3.6.
The mean of the binomial distribution is np, and the variance of the binomial distribution is np (1 − p). When p = 0.5, the distribution is symmetric around the mean—such as when flipping a coin because the chances of getting heads or tails is 50%, or 0.5. When p > 0.5, the distribution curve is skewed to the left. When p < 0.5, the distribution curve is skewed to the right.