91Ó°ÊÓ

To solve problems related to real-world events like the commissioner's claim, we often use probability calculations. In this exercise, we're dealing with a _binomial distribution_.
The binomial distribution is ideal for situations where there are two possible outcomes, like support or no support. We can specify a probability of an event occurring, noted as \(p\). In this case, \(p = 0.8\), indicating that the claimed proportion of the city's support for private garbage collection is 80%.
The first step is to determine the probability of observing at least 22 out of 25 supporters. To find this, we sum up the probabilities of observing exactly 22, 23, 24, and 25 supporters using:

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

where \( \binom{n}{k} \) is a binomial coefficient, showing the number of ways to choose \(k\) successes from \(n\) trials. By computing and adding these probabilities, we get the overall probability for the desired event.

Statistical inference helps us make conclusions about a population based on sample data. In this scenario, we're using the sample of 25 people to infer about the entire city's opinion on private garbage collection.
In our exercise, we aim to determine if the observed data, where 22 out of 25 favor the proposal, aligns with the commissioner's claim of 80% support. Through probability calculation, a core aspect of statistical inference, we can evaluate how probable it is to observe such results if the true proportion is 0.8.
The calculated probability gives us an idea of how likely the observed sample results would occur under given assumptions.

• If the probability for the sample matches or exceeds a certain threshold (e.g., 0.05), it suggests the sample is consistent with the population parameter p = 0.8.
• If it's less, we may infer that the claim may not be accurate.

Thus, statistical inference enables us to make educated assumptions or reject claims about a larger group based on sample data.

Hypothesis Testing is a systematic method used to evaluate claims or hypotheses about a population. In our example, the hypothesis in question is that 80% of the city supports privatization of garbage collection.
Here's how hypothesis testing works in this context:

• **Null Hypothesis \((H_0)\)**: The proportion of support is 80%, or \( p = 0.8 \).
• **Alternative Hypothesis \((H_a)\)**: The proportion of support is not 80%, \( p eq 0.8 \).

We then use our sample data and calculate the probability of observing a result as extreme as, or more extreme than, the actual sample (in this case, 22 supporters). This probability is central in determining if our null hypothesis can hold.
If the calculated probability is less than a predetermined significance level (often 0.05), there is sufficient evidence to reject the null hypothesis, suggesting the population proportion differs from 80%.
Conversely, if the probability is higher, we fail to reject the null hypothesis, implying there's not enough evidence to dispute the commissioner's claim.