91Ó°ÊÓ

The term **point estimate** refers to a single value that serves as a best guess or approximation for an unknown population parameter. In the context of our exercise, the point estimate for the population proportion, denoted as \( \hat{p} \), is the observed proportion of inaccurate orders in our sample data. To find this, we divide the number of inaccurate orders by the total number of orders observed. This gives us the formula:
\[ \hat{p} = \frac{33}{362} \approx 0.0912 \]
Here, \(0.0912 \) or **9.12%** is our point estimate, suggesting that approximately 9.12% of McDonald’s orders in our sample were not accurate.
By using a point estimate, we simplify our understanding of a population's characteristics based on sample data. However, it is important to remember that it does not account for uncertainty or variability—this is where the margin of error and confidence intervals come into play.

The **margin of error (E)** accounts for the variability in sampling and provides a range within which the true population parameter is expected to lie with a certain level of confidence.
To calculate the margin of error for a population proportion, we use the formula:
\[ E = Z_{\alpha/2} \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \]
In this formula:

• \( Z_{\alpha/2} \) is the z-score corresponding to our confidence level (1.96 for a 95% confidence level).
• \( \hat{p} \approx 0.0912 \) is our point estimate of the population proportion.
• \( n = 362 \) is the total number of observed orders.

Substituting the values, we get:
\[ E = 1.96 \sqrt{\frac{0.0912(1 - 0.0912)}{362}} \approx 0.029 \]
The margin of error in this case is approximately **0.029** or **2.9%**. This means that the true proportion of inaccurate orders could be as much as 2.9% higher or lower than our point estimate of 9.12%.

A **population proportion** is a measure that describes the fraction of the population that displays a certain characteristic. In our exercise, we are interested in the proportion of all McDonald’s drive-through orders that are inaccurate. Our goal is to estimate this proportion based on sample data.
The sample proportion calculated earlier (\( \hat{p} = 0.0912 \)) serves as an estimate for the population proportion. However, due to sampling variability, it’s unlikely that the sample proportion equals the population proportion exactly.
This is why constructing a confidence interval is useful; it provides a range of values within which the true population proportion is likely to fall. A confidence interval offers more information than a single point estimate by indicating the reliability of the estimate and the degree of uncertainty associated with it.

A **95% confidence level** indicates that if we were to take 100 different samples and compute a confidence interval for each, we would expect about 95 of those intervals to contain the true population proportion.
To compute the confidence interval, we use the point estimate and margin of error to create the range:
\[ \hat{p} - E \leq p \leq \hat{p} + E \]
Plugging in our values:
\[ 0.0912 - 0.029 \leq p \leq 0.0912 + 0.029 \]
This simplifies to:
\[ 0.0622 \leq p \leq 0.1202 \]
Therefore, we are **95% confident** that the true proportion of inaccurate orders at McDonald’s is between **6.22%** and **12.02%**.
This interpretation means there is high confidence that the actual error rate lies within this range, allowing us to account for the uncertainty inherent in sample data.