Problem 20 - A First Course in Probability

Question

If 65 percent of the population of a large community is in favor of a proposed rise in school taxes, approximate the probability that a random sample of 100 people will contain (a) at least 50 who are in favor of the proposition; (b) between 60 and 70 inclusive who are in favor; (c) fewer than 75 in favor.

Step-by-Step Solution

Answer

The approximate probabilities for each scenario are as follows: (a) At least 50 people in favor: \(P(X ≥ 50) ≈ 0.9992\) (b) Between 60 and 70 people inclusive in favor: \(P(60≤X≤70) ≈ 0.7062\) (c) Fewer than 75 people in favor: \(P(X < 75) ≈ 0.9826\)

Step 1: Find the population proportion (p), and the standard deviation of the sample proportion

The given population proportion (p) is 65%, or 0.65. Since it is a binomial distribution, we can find the standard deviation of the sample proportion (σ) using the formula: \(σ = \sqrt{\frac{p(1-p)}{n}}\) where n is the sample size, which is 100 in this case. Plug in the numbers and we get: \(σ = \sqrt{\frac{0.65(1-0.65)}{100}} ≈ 0.0475\)

Step 2: Use the Z-score formula for each scenario

We will use the Z-score formula to find the probabilities for each scenario. The Z-score formula is: \(Z = \frac{\hat{p} - p}{σ}\) where \(\hat{p}\) is the sample proportion.

Step 3: Calculate the probability for scenario (a)

(a) At least 50 people are in favor of the proposition: To find the probability of at least 50 people in favor, we first find the Z-score for 50 people, or \(p_a = 0.50\). \(Z_a = \frac{0.50 - 0.65}{0.0475} ≈ -3.16\) Using a Z-table or a calculator, the probability for a Z-score of -3.16 is approximately 0.0008. Since we want the probability of at least 50 people in favor, we need to find the complement of this probability. \(P(X ≥ 50) = 1 - P(X<50) = 1 - 0.0008 = 0.9992\) The probability of a random sample of 100 people containing at least 50 in favor is approximately 0.9992.

Step 4: Calculate the probability for scenario (b)

(b) Between 60 and 70 people inclusive are in favor: To find the probability for this scenario, we need to find the Z-scores for 60 and 70 people, or \(p_b1 = 0.60\) and \(p_b2 = 0.70\). \(Z_{b1} = \frac{0.60 - 0.65}{0.0475} ≈ -1.05\) \(Z_{b2} = \frac{0.70 - 0.65}{0.0475} ≈ 1.05\) Using the Z-table or a calculator, the probabilities for Z-scores of -1.05 and 1.05 are approximately 0.1469 and 0.8531. The probability of a random sample of 100 people containing between 60 and 70 people in favor is: \(P(60≤X≤70) = P(Z_{b2}) - P(Z_{b1}) = 0.8531 - 0.1469 = 0.7062\)

Step 5: Calculate the probability for scenario (c)

(c) Fewer than 75 people are in favor: To find the probability of fewer than 75 people in favor, we first find the Z-score for 75 people, or \(p_c = 0.75\). \(Z_c = \frac{0.75 - 0.65}{0.0475} ≈ 2.11\) Using a Z-table or a calculator, the probability for a Z-score of 2.11 is approximately 0.9826. Since we want the probability of fewer than 75 people in favor, we don't need to find the complement of this probability. \(P(X < 75) = 0.9826\) The probability of a random sample of 100 people containing fewer than 75 in favor is approximately 0.9826.

Key Concepts

Binomial Distribution

Understanding the binomial distribution is key when dealing with probability questions where there are two possible outcomes, such as 'success' or 'failure'. In our example, the success is individuals in favor of a tax proposition. The binomial distribution requires a fixed number of trials, independent events, and a constant probability for each trial. The formulas associated with it allow us to calculate probabilities and various statistics about the distribution of successes.

With the binomial distribution, we can answer questions like the probability of a certain number of successes in a sample. This is achieved by using the binomial formula, which incorporates the sample size, the probability of success, and the number of successes of interest.

Sample Proportion

The sample proportion, often denoted as \( \hat{p} \), represents the proportion of success within a sample. For instance, if you're sampling 100 people for their opinion on a policy, and 60 support it, the sample proportion would be \( \hat{p} = 0.60 \). In probability and statistics, the sample proportion helps us estimate the population proportion, and this estimation becomes more accurate with larger sample sizes. Calculating sample proportions allows us to make hypotheses and predictions about the larger population, which in our exercise involves estimating support for a tax rise.

Standard Deviation

The standard deviation, symbolized by \( \sigma \) or 's', quantifies the dispersion or variation of a set of data points. In the context of a binomial distribution, it helps us understand how much the sample proportion is likely to vary from the population proportion. The standard deviation of the sample proportion, specifically, is found using the formula \( \sigma = \sqrt{\frac{p(1-p)}{n}} \), where 'p' is the population proportion and 'n' is the sample size. A smaller standard deviation indicates that data points tend to be closer to the mean; larger values suggest a wider spread.

Z-score

A Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is measured in terms of standard deviations from the mean. The Z-score formula is \( Z = \frac{\hat{p} - p}{\sigma} \), where \( \hat{p} \) is the sample proportion and \( \sigma \) is the standard deviation of the sample proportion. Z-scores allow us to calculate the probability of a score occurring within a normal distribution and enable us to compare two scores from different distributions. In our exercise, we can use the Z-score to determine the likelihood of seeing at least 50, between 60 and 70, or less than 75 individuals in favor of the tax rise in a sample of 100 people.

Population Proportion

The population proportion, denoted as 'p', is the actual proportion of individuals in a population who have a particular characteristic—in our exercise, this is the proportion that favors the tax increase. The given population proportion is an essential component in calculating the standard deviation as well as the expected value of the sample proportion. It serves as the benchmark when determining the probability of observing a certain sample proportion, like finding out what portion of a population sample supports a proposition. When paired with data gathered from the sample (like a survey of 100 people), we can use it to make inferences about the larger community's sentiments.