Question

Exercise 1 Exit polling has been a controversial practice in recent elections, since early release of the resulting information appears to affect whether or not those who have not yet voted do so. Suppose that 90% of all registered Wisconsin voters favor banning the release of information from exit polls in presidential elections until after the polls in Wisconsin close. A random sample of 250 Wisconsin voters are selected (You can assume that the responses of those surveyed are independent). Let X be the count of people in the 250 who favor the ban.

a. Calculate the probability that exactly 230 people in the sample of 250 favor the ban, that is P(X = 230). b. Calculate the exact probability that 230 or more people in the sample of 250 favor the ban,
that is P(X230). Hint: use a couple of R. functions to help with this calculation.
c. What are the expected value (x) and standard deviation (ox) of X?
d. We can consider X the sum of 250 iid random draws from the population Y where P(Y=1)=0.90 and P(Y=0)=0.10. That is X = Y + Y₂++Y250. What do we think will be true about the shape of the distribution of X? What theorem are you using?
e. To look at how well a normal curve approximates the distribution of X with n=250, = 0.90 run the following code with the mean and sd values computed in (c) substituted for "MEAN VALUE" and "SD_VALUE", respectively. You will also need to update eval=TRUE.
plot(x-seq(150,250,1), y=dbinom (150:250, 250, prob-0.90), type='h', ylab="") curve (dnorm(x, mean-MEAN_VALUE, sd-SD_VALUE), col="darkblue", lwd-2, add-TRUE, yaxt="n")
f. Calculate the approximate probability that at least 230 people in a sample of 250 favor the ban, that is P(X > 230), assuming a Normal Distribution for X centered at the mean and sd found in c. Compare the value to that found in b. and explain why they are not exactly equal.1
g. Consider the sample proportion. Calculate the approximate probability that at least p = 230 favor the ban, that is P(p > 0.92), assuming a Normal Distribution of p centered at the appropriate mean and standard deviation. Compare the value to that found in f. and explain the relationship between the values.

Answer 1

This mathematics question entails calculating probabilities using binomial and normal distributions, understanding the expected value and standard deviation for a binomially distributed random variable, applying the Central Limit Theorem, and comparing approximations from normal distributions to exact binomial calculations.

Step-by-step explanation:

Exercise 1: Understanding Binomial Distribution and Normal Approximation

a. The probability that exactly 230 people favor the ban is calculated using the binomial probability formula P(X = k) = C(n, k) * p^k * (1-p)^(n-k), where p is the probability of a single success, n is the number of trials, k is the number of successes, and C(n, k) is the combination of n items taken k at a time.

b. To find the probability that 230 or more people in the sample favor the ban, one would sum the probabilities of all outcomes from 230 to 250, which can be done efficiently using cumulative distribution functions in statistical software.

c. The expected value (mean), μ, of X is calculated as n*p, and the standard deviation, σ, is calculated as sqrt(n*p*(1-p)).

d. Due to the large sample size and high probability of success, we can expect X to have an approximately normal distribution, according to the Central Limit Theorem.

e. The R code mentioned is used to compare the actual binomial distribution of X to its normal approximation graphically.

f. To approximate P(X ≥ 230) assuming X is normally distributed, one would use the standard normal distribution and z-scores. The approximation may differ slightly from the exact value due to the discrete nature of the binomial distribution versus the continuous normal distribution.

g. For the sample proportion, π, P(π > 0.92) can be estimated using a normal distribution for the sample proportion, with a mean of p and a standard deviation of sqrt(p*(1-p)/n). This value will be related to but not identical to that found in part f, due to differences in the scaling factor for proportions versus counts.