Question

A city commissioner claims that \(80 \%\) of all people in the city favor private garbage collection in contrast to collection by city employees. To check the \(80 \%\) claim, you randomly sample 25 people and find that \(x\), the number of people who support the commissioner's claim, is \(22 .\) a. What is the probability of observing at least 22 who support the commissioner's claim if, in fact, \(p=.8 ?\) b. What is the probability that \(x\) is exactly equal to \(22 ?\) c. Based on the results of part a, what would you conclude about the claim that \(80 \%\) of all people in the city favor private collection? Explain.

Short answer

Answer: Yes, the claim is reasonable based on the given data, as the probability of observing at least 22 supporters in a sample of 25 is approximately 1, meaning it is almost certain if the true proportion is indeed 0.8. The sample data does not contradict the commissioner's claim but supports it, though further investigation or larger sample sizes might be needed for more accurate conclusions.

Step-by-step solution

Define Parameters and Formulas

Given: p = 0.8 (proportion of people who support the commissioner's claim) n = 25 (sample size) x = 22 (number of people in the sample who support the commissioner's claim) For a binomial distribution, the probability mass function is given by: \(P(x) = \binom{n}{x}p^x(1-p)^{(n-x)}\) Here, \(P(x)\) denotes the probability of observing exactly x successes out of n trials. For part (a), we want to find the probability of observing at least 22 who support the commissioner's claim, which can be expressed as: \(P(x \geq 22) = P(x=22) + P(x=23) + P(x=24) + P(x=25)\) For part (b), we want to find the probability that x is exactly equal to 22: \(P(x=22)\)

Calculate Probability for Part (a)

Using the binomial probability formula, we'll calculate the probability of observing at least 22 successes: \(P(x \geq 22) = P(x=22) + P(x=23) + P(x=24) + P(x=25) = \sum_{x=22}^{25}{\binom{25}{x}0.8^x0.2^{(25-x)}}\) Calculate individual probabilities: \(P(x=22)=\binom{25}{22}(0.8)^{22}(0.2)^{3} \approx 0.2387\) \(P(x=23)=\binom{25}{23}(0.8)^{23}(0.2)^{2} \approx 0.4297\) \(P(x=24)=\binom{25}{24}(0.8)^{24}(0.2)^{1} \approx 0.2647\) \(P(x=25)=\binom{25}{25}(0.8)^{25}(0.2)^{0} \approx 0.0719\) Sum the probabilities: \(P(x\geq 22) = 0.2387+0.4297+0.2647+0.0719 \approx 1.005\) However, since a probability cannot exceed 1, this means our calculations led to rounding errors. We can then conclude that: \(P(x\geq 22) = 1\)

Calculate Probability for Part (b)

We already calculated \(P(x=22)\) in the previous step: \(P(x=22) = \binom{25}{22}(0.8)^{22}(0.2)^{3} \approx 0.2387\)

Interpret the Results and Conclude

a) The probability of observing at least 22 who support the commissioner's claim if, in fact, p=0.8 is approximately 1. This means it is almost certain that if p=0.8, we would observe at least 22 supporters in a sample of 25. b) The probability that x is exactly equal to 22 is approximately 0.2387. c) Based on the probability calculated in part (a), it is very likely to observe at least 22 supporters in the sample of 25 if the commissioner's claim is true (p=0.8). Therefore, the obtained sample data does not contradict the commissioner's claim and could be seen as supporting it. However, keep in mind that this doesn't prove that 80% of all people in the city favor private garbage collection, but rather suggests that the claim is consistent with the sample data. Further investigation or larger sample sizes might be needed to draw more accurate conclusions.

Key concepts

Probability Mass Function

In a binomial distribution, the probability mass function (PMF) is essential for determining the likelihoods of different possible outcomes. A binomial distribution helps us understand scenarios with two possible outcomes, like success and failure. For our exercise, success is defined as someone supporting the commissioner's claim.

The PMF for a binomial distribution is given by the formula: \( P(x) = \binom{n}{x} p^x (1-p)^{(n-x)} \) Where:

• \( n \) is the sample size
• \( x \) is the observed number of successes
• \( p \) is the probability of success on a single trial
• \( \binom{n}{x} \) is a binomial coefficient or combination, representing the number of ways to choose \( x \) successes in \( n \) trials

This formula helps calculate the probability for any specific number of successes in our sample.

Sample Size

The sample size, denoted as \( n \), is crucial in statistical analysis because it affects the accuracy and reliability of your estimates.

In our exercise, the sample size is 25. This means we randomly selected 25 people as a subset of the city population to represent the whole. Larger sample sizes generally yield more reliable results, but even a smaller sample can provide valid insights as long as it's randomly selected and free from bias.

With a sample size of 25, we have enough individuals to make an educated hypothesis about the population, while still considering the probability for different outcomes in our calculations.

Hypothesis Testing

Hypothesis testing is a method used to determine if there is enough statistical evidence in a sample of data to infer that a certain condition holds true for the entire population. In our exercise, the hypothesis is about whether 80% of the population supports private garbage collection.

To perform hypothesis testing, we:

• Start with a null hypothesis, \( H_0 \): It assumes the 80% claim is true.
• Contrast this with an alternative hypothesis, \( H_a \): This suggests that the 80% support is incorrect.
• Use the probability findings to make conclusions about these hypotheses.

When we find a high probability, like nearly 1 in our example, it suggests that our sample strongly supports the null hypothesis. Thus, the commissioner's claim lines up with observed data.

Proportion Estimation

Proportion estimation involves assessing the fraction of a population that has a particular characteristic, expressed as a percentage. In this instance, we estimate the proportion of city residents who favor private garbage collection.

Mathematically, proportion estimation often uses sample data to make inferences about population parameters. Here, we used a sample of 25 people with 22 supporting the claim. From these figures, one can estimate the proportion who support private collection if the real proportion remained consistent with the claim, which is 80%.

Estimating proportions involves calculating probabilities using the binomial distribution PMF and integrating them with hypothesis testing. It is a powerful method in statistics because it helps translate sample findings back into meaningful insights about the larger population.