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Probability calculation when dealing with a binomial distribution involves finding the likelihood of a given number of "successes" in a series of independent trials. In this exercise, we are analyzing a situation where residents of a city preferred a private garbage collection company. We want to calculate specific probabilities based on the commissioner's claim that 80% of people prefer this option.

### Understanding the ParametersRecognize this as a binomial experiment. Here:

• Number of trials () is 25, representing the sampled residents.
• Probability of success (p) is 0.8, as per the commissioner's claim.

Next, use these to calculate probabilities for specific scenarios, such as having exactly or at least a certain number of residents indicating a preference for private contracting.

### Calculating Specific ProbabilitiesTwo primary formulas are at play:

• Exact Probability Formula: \[ P(X = k) = {\binom{n}{k}} p^k (1-p)^{n-k} \] calculates the probability of exactly k successes.
• Cumulative Probability Formula: \[ P(X \geq k) = 1 - P(X \leq k-1) \] helps find the probability of getting at least k successes.

Apply these tools to see how "at least 22" or "exactly 22" preferences manifest in terms of probabilities.

Statistical claims testing helps us verify or challenge claims made about a population. In this case, the commissioner's assertion that 80% of residents favor a private garbage company is under scrutiny. The aim is to test this claim using sample data and statistical tools.

### Hypothesis Testing Basics To understand if the commissioner's statement holds:

• Null Hypothesis (H0): This assumes the claim is true, i.e., 80% prefer privatization.
• Alternative Hypothesis (H1): This suggests the claim is false, indicating a different preference rate.

Using the binomial distribution and accumulated sample data, calculate the relevant probabilities to find if the observed data (22 out of 25) aligns well with the commissioner's claim.

### Evaluating Outcomes If the calculated probability, that at least 22 rather than 80% prefer privatization, turns out significantly low, it might indicate the claim is not true. The conclusion depends on the significance level (often 5% ) and whether the observed data lies within typical expectations of the binomial distribution.

Cumulative probability is a powerful concept in probability theory, especially useful when dealing with the binomial distribution. It represents the probability that a random variable takes on a value less than or equal to a specific number, integrating over all possible outcomes at once.

### Computing Cumulative ProbabilityTo find cumulative probabilities using a binomial distribution, consider calculating \( P(X \leq k) \), the sum of the probabilities of all possible success counts up to \( k \) successes:\[ P(X \leq k) = \sum_{x=0}^{k} {\binom{n}{x}} p^x (1-p)^{n-x} \]This calculation often forms the basis for obtaining probabilities such as \( P(X \geq 22) \) by subtracting the cumulative probability through a complement approach:

\[ P(X \geq k) = 1 - P(X \leq k-1) \]

### Application in Testing ClaimsBy evaluating cumulative probabilities, assess how rare the observed outcome (22 or more successes in our scenario) is. It's crucial in verifying if deviations from expected outcomes (under the claim) could be attributed to random chance or if they suggest discrepancies in the claim itself.