91Ó°ÊÓ

Probability helps us understand how likely it is that a certain event will occur. In the context of the exit poll, we want to know the likelihood that all 1751 voters support Charles Schumer given that 65% of the overall population does so. In this case, the binomial distribution is the tool we use. It's perfect for scenarios that involve a fixed number of independent trials, like casting a vote.

The probability formula for a binomial distribution is:

• \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \)

Here, \( n \) is the number of trials (voters), \( k \) is the exact number of successes we care about (all voters choosing Schumer), and \( p \) is the probability of success on each trial (the chance a single voter supports Schumer).

For the scenario given, \( n = 1751 \) and \( k = 1751 \), which means we're calculating the probability that every single one of the 1751 voters picked Schumer. Plug in the respective values to find:

• \( P(X = 1751) = (0.65)^{1751} \)

This probability is so tiny, it is almost zero, emphasizing that such an outcome is nearly impossible.

Standard deviation measures the amount of variation or dispersion in a set of values. When working with proportions, standard deviation tells us how much the sample proportion could fluctuate around the population proportion. In the exit poll example, we use it to see how much the sample proportion of those who support Schumer (hypothetically 1.0) deviates from the known population proportion of 0.65.

The formula to calculate the standard deviation of a sample proportion is:

• \( \sigma = \sqrt{\frac{p(1-p)}{n}} \)

Where \( p \) is the population proportion and \( n \) is the sample size (1751 voters).

By substituting \( p = 0.65 \) and \( n = 1751 \), we can compute the standard deviation. This calculation helps to understand the expected range of sample proportions due to random sampling. Knowing the standard deviation prepares us to assess unusual results effectively.

A Z-score quantifies how far away a particular measurement is from the mean, in terms of standard deviations. In the case of the exit poll, it indicates how extreme having all 1751 voters support Schumer is relative to what is expected, based on the population proportion.

The formula for the Z-score is:

• \( Z = \frac{\hat{p} - p}{\sigma} \)

In this formula, \( \hat{p} = 1.0 \) is the sample proportion of voters for Schumer, \( p = 0.65 \) is the expected proportion, and \( \sigma \) is the standard deviation we computed earlier.

Calculating the Z-score with these values gives us an insight into how unusual the result is. A high Z-score means that all voters supporting Schumer is far from the expected proportion, highlighting the rarity and significance of such an outcome.