Final Answer:
The mean of the sampling distribution for the proportion of supporters with sample size n = 155 is 0.48.
Step-by-step explanation:
The population proportion of voters in the county who support a new fire district is denoted by p. The sampling distribution of p is a distribution of all possible values of the sample proportion, calculated from random samples of size n = 155. The mean of the sampling distribution of the proportion of supporters is equal to the population proportion p. Since the population proportion is given as 0.48, the mean of the sampling distribution is also 0.48.
The mean of the sampling distribution can also be calculated using the formula μ = np, where μ is the mean of the sampling distribution, n is the sample size, and p is the population proportion. In this case, μ = 155 × 0.48 = 74.4, which is the same as the population proportion of 0.48.
The sampling distribution of p follows a normal distribution curve, with the mean equal to the population proportion. The mean of the sampling distribution is the most likely sample proportion from the sampling distribution. The proportion of supporters in any given sample of size 155 is most likely to be 0.48.
The standard deviation of the sampling distribution is calculated using the formula σ = √[np(1-p)]. In this case, σ = √[155 × 0.48 × (1-0.48)] = 0.0459. The standard deviation gives an indication of how close the sample proportions are likely to be to the population proportion of 0.48. Most samples of size 155 are likely to have a proportion of supporters within 0.0459 of the population proportion.
In conclusion, the mean of the sampling distribution for the proportion of supporters with sample size n = 155 is 0.48. This is equal to the population proportion. The standard deviation of the sampling distribution is 0.0459, which indicates how close the sample proportions are likely to be to the population proportion of 0.48.