Final answer:
a. The approximate distribution of X is a binomial distribution. b. 1,956.8 is a population mean. c. To find the probability that a randomly selected district had fewer than 1,600 votes for the candidate, we use the normal distribution. To find the probability that a randomly selected district had between 1,800 and 2,000 votes for the candidate, we subtract the cumulative probability up to 1,800 from the cumulative probability up to 2,000. To find the third quartile for votes for the candidate, we find the value of X for which the cumulative probability is 0.75.
Step-by-step explanation:
a. The approximate distribution of X is a binomial distribution.
In this case, X represents the number of Democrats out of the 500 residents responding to the poll.
The binomial distribution is appropriate because we have a fixed number of trials (500) and each trial has two possible outcomes (Democrat or not Democrat).
b. 1,956.8 is a population mean. We know this because it is not based on data from a sample, but rather it represents a value for the entire population.
c. To find the probability that a randomly selected district had fewer than 1,600 votes for the candidate, we need to determine the cumulative probability up to 1,600.
This can be calculated using the normal distribution with the given mean and standard deviation. We can then sketch the graph and write the probability statement as P(X < 1600).
To find the probability that a randomly selected district had between 1,800 and 2,000 votes for the candidate, we need to determine the cumulative probability up to 2,000 and subtract the cumulative probability up to 1,800.
This can also be calculated using the normal distribution with the given mean and standard deviation.
The probability statement can be written as P(1800 < X < 2000).