Problem 38 - An Introduction to Mathematical Statistics and Its Applications

Question

Fifty-five percent of the registered voters in Sheridanville favor their incumbent mayor in her bid for re-election. If four hundred voters go to the polls, approximate the probability that (a) the race ends in a tie. (b) the challenger scores an upset victory

Step-by-Step Solution

Answer

(a) The approximate probability that the race ends in a tie is represented by the Z-score probability found for \(Z_1\). (b) The approximate probability that the challenger scores an upset victory is represented by the Z-score probability found for \(Z_2\).

Step 1: Identify Variables

The probability of success, \(p\), is the percentage favoring the incumbent mayor converted into decimal ie. \(0.55\). The number of trials, \(n\), is the number of voters ie. \(400\). The number of successful outcomes, \(x\), will be for (a) half of \(400\) ie. \(200\) and for (b) less than \(200\). Also, calculate the mean, \(μ = np = 400 * 0.55 = 220\) and the standard deviation, \(σ = \sqrt{np(1-p)} = \sqrt{400*0.55*(1-0.55)}\).

Step 2: Calculate Z-Scores

Use the formula for Z-Score to standardize the random variable X. For (a) \(Z_1 = (X - μ)/σ = (200 - 220)/σ\) and for (b) \(Z_2 = (199 - 220)/σ\), considering continuity correction.

Step 3: Calculate Probability Using Z-Scores

The values of Z-Scores obtained can be used to find probabilities from a standard normal table. For (a) the probability of tie is \[P(Z > -Z_1)\] and for (b) the probability of upset victory is \[P(Z < -Z_2)\]. These will give the desired probabilities.

Step 4: Calculate Z-Score Probabilities

For step 3, use a standard normal distribution table to find the probabilities related to the calculated Z-scores.

Key Concepts

Statistics

Statistics is a branch of mathematics dealing with data collection, analysis, interpretation, and presentation. It is essential in making informed decisions based on data.
In the context of our exercise, we look at votes as data points to understand election outcomes. Each voter's choice can be seen as a success or failure.

• The probability of a voter supporting the incumbent mayor is given as 55% or 0.55.
• The total number of voters, i.e., trials, is 400, which is our sample size.
• An essential part of statistics is interpreting these numbers to calculate probabilities and outcomes.

Statistics helps in making predictions, like approximating whether an election race ends in a tie or if the challenger wins. Observing patterns and performing statistical testing allows us to determine if an outcome is likely or unusual. By understanding key concepts like mean and standard deviation, we can delve deeper into probability calculations.

Probability Distribution

Probability distributions describe how the values of a random variable are distributed. For binary or binomial scenarios like an election, where every voter has two outcomes (support or not support the mayor), a binomial probability distribution is employed.
In our scenario, the incumbent mayor's support is a success, quantified by a probability of 0.55 for each of the 400 voters.

• The mean of this distribution is calculated as \( \, μ = np = 400 \times 0.55 = 220 \).
• The standard deviation tells us how much variability there is from the mean: \( \, σ = \sqrt{np(1-p)} = \sqrt{400 \times 0.55 \times (1-0.55)} \).
• The binomial distribution is useful for calculating exact probabilities for a number of successes, like the mayor getting exactly 200 or more votes.

By understanding how probability distributions work, we gain insights into different possible outcomes and their associated probabilities in the election scenario. It helps quantify and visualize occurrences like a tie or an upset and determines their likelihood.

Voting Models

Voting models use mathematical frameworks to predict the outcome of elections. These models consider probabilities and other statistical measures to assess potentials like wins or losses.
In our example, the voting model relies on a binomial probability distribution to forecast outcomes among 400 voters. For a fair prediction:

• The probability of each voter supporting the incumbent is 0.55.
• The election race ending in a tie or a challenger victory are calculated using these probabilities.
• By evaluating different scenarios using statistical measures, we can model various electoral outcomes and their probabilities.

Voting models, therefore, provide us with critical insights into how a particular election might unfold. They form the basis for strategizing campaigns, understanding voter dynamics, and even policy-making.

Z-Score Calculation

Z-Score calculation is a statistical technique used to convert a data point to a standard normal distribution scale. It shows how many standard deviations a point is from the mean. In our context, Z-scores help us gauge the likelihood of specific election outcomes.

• To calculate the Z-score for a tie (200 votes): \( Z_1 = \frac{200 - 220}{σ} \).
• For an upset victory (less than 200 votes), considering slight adjustment: \( Z_2 = \frac{199 - 220}{σ} \).
• These Z-scores allow us to refer to a standard normal distribution table to find respective probabilities.

Incorporating Z-score calculations aids in understanding how far each scenario is from the norm. It simplifies comparing different outcomes against a standard and enables us to confirm whether an outcome is within the expected range or notably different. This statistical method is foundational in predicting probable vote shares and assessing election scenarios.