Answer :
To determine the variance of the proportion of people surveyed who say they support the incumbent candidate, follow these steps:
Understand the Problem: You have a population consisting of an equal number of men and women. Men have a 70% support rate for the incumbent, while women have a 30% support rate. The sample size for the survey is 500 people.
Calculate the Proportion of Supporters in the Population:
Since the population is evenly split between men and women, let's assume there are n men and n women in the population.
- Proportion of male supporters = 0.7
- Proportion of female supporters = 0.3
The overall proportion of supporters in the population is the average of these two proportions:
[tex]p = \frac{0.7n + 0.3n}{2n} = \frac{0.7 + 0.3}{2} = 0.5[/tex]
Calculate Variance of the Proportion:
The variance of a proportion in a sample can be given by the formula:
[tex]\text{Variance} = \frac{p(1 - p)}{n}[/tex]
where [tex]p[/tex] is the proportion of supporters, and [tex]n[/tex] is the total number of people surveyed.
Plugging in the values, we get:
[tex]\text{Variance} = \frac{0.5(1 - 0.5)}{500}[/tex]
[tex]\text{Variance} = \frac{0.5 \times 0.5}{500} = \frac{0.25}{500} = 0.0005[/tex]
Therefore, the variance of the proportion of people who support the incumbent in the survey is 0.0005.
Keep in mind that this variance refers to the variability in the sample proportion of people who support the incumbent, considering a simple random sample.