Final answer:
To approximate the mean of a frequency distribution, you multiply the midpoint of each group by its frequency, sum these products, and divide by the total number of observations. Probabilities and quartiles are also calculated using the distribution's properties visualized in a histogram.
Step-by-step explanation:
Approximating the mean of the frequency distribution for populations involves using grouped data to estimate the actual mean. The process requires taking the midpoint of each group, multiplying it by the frequency of the group, summing these products, and then dividing by the total number of observations. When calculating the mean, sample mean is used when the data comes from a subset of the population, while population mean refers to data from the entire population. Since we are considering the counties in a state, it is likely we are looking at a population mean, although this can depend on whether the data represents all counties or just a sample.
The probability of a randomly selected district having fewer than 1,600 votes for the candidate depends on the distribution of the data, which can be visualized using a histogram and corresponding probabilities. Similarly, finding the third quartile for votes requires analyzing the frequency distribution and identifying the value below which 75% of the data falls. The smooth curve drawn over the histogram will give an indication of the distribution's shape, which is essential for probability calculations and for estimating measures like quartiles.