In the 1992 presidential election, Alaska's 40 election districts averaged 2050 votes per district for President Clinton. The standard deviation was 559. (There are only 40 election districts in Alaska.) The distribution of the votes per district for President Clinton was bell-shaped. Let $X=$ number of votes for President Clinton for an election district. (Source: The World Almanac and Book of Facts) Round all answers except part e. to 4 decimal places.
a. What is the distribution of $\mathrm{X}$ ? $\mathrm{X} \sim \mathrm{N}($
b. Is 2050 a population mean or a sample mean? Select an answer $v$
c. Find the probability that a randomly selected district had fewer than 1884 votes for President Clinton.
d. Find the probability that a randomly selected district had between 2087 and 2245 votes for President Clinton.
e. Find the third quartile for votes for President Clinton. Round your answer to the nearest whole number.
Hint:
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Answer
The third quartile for votes for President Clinton is approximately \(\boxed{2427}\). This is calculated using the inverse of the CDF (also known as the quantile function) at 0.75.
Step 1 ：The distribution of X is given by \(X \sim N(2050, 559^2)\). This is because the number of votes for President Clinton for an election district, denoted by X, follows a normal distribution with mean 2050 and standard deviation 559.
Step 2 ：The value 2050 is a population mean. This is because we have data for all 40 election districts in Alaska, so 2050 represents the average number of votes for President Clinton across the entire population of districts.
Step 3 ：The probability that a randomly selected district had fewer than 1884 votes for President Clinton is approximately \(\boxed{0.3832}\). This is calculated using the cumulative distribution function (CDF) of the normal distribution.
Step 4 ：The probability that a randomly selected district had between 2087 and 2245 votes for President Clinton is approximately \(\boxed{0.1100}\). This is calculated by finding the difference between the CDF at 2245 and the CDF at 2087.
Step 5 ：The third quartile for votes for President Clinton is approximately \(\boxed{2427}\). This is calculated using the inverse of the CDF (also known as the quantile function) at 0.75.
