91Ó°ÊÓ

In the context of the given exercise, the sample proportion is a fundamental concept to understand. The sample proportion, denoted as \( \hat{p} \), represents the proportion of inaccurate orders in your sample. You calculate it by dividing the number of inaccurate orders (\( x \)) by the total number of orders (\( n \)).

Here's the formula again for better clarity: \[ \hat{p} = \frac{x}{n} \]
So, for this exercise: \[ \hat{p} = \frac{33}{362} \approx 0.0912 \]
Think of the sample proportion as a point estimate that gives us an idea of what the true population proportion might be. It's crucial because all further calculations, such as the standard error and confidence interval, hinge on this value. This number answers the question, 'Out of all the orders we checked, how many were inaccurate?'

The standard error (SE) helps us understand the variability or uncertainty in our sample proportion. In simpler terms, it tells us how much we can expect our sample proportion to vary if we were to take many different samples from the same population.

The formula for standard error of the sample proportion is: \[ SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \]
Here, \( \hat{p} \) is the sample proportion, and \( n \) is the total number of observations in your sample.

For this exercise, we have: \[ SE = \sqrt{\frac{0.0912(1-0.0912)}{362}} \approx 0.0147 \]
A smaller standard error means the sample proportion is a more precise estimate of the population proportion. Conversely, a larger SE indicates greater uncertainty and variability.

In constructing confidence intervals, the Z-score (\( Z^* \)) plays a critical role. The Z-score corresponds to the desired confidence level of your interval. For a 95% confidence level, the Z-score is 1.96, meaning that 95% of the sample proportions would fall within this range if you were to take many samples.

To construct the confidence interval, you need to calculate the margin of error (ME) using the formula: \[ ME = Z^* \times SE \]
Using the values from the exercise: \[ ME = 1.96 \times 0.0147 \approx 0.0288 \]
Finally, you use this margin of error to create the confidence interval: \[ CI = \hat{p} - ME \text{ to } \hat{p} + ME \]
Substituting the calculated values: \[ CI = 0.0912 - 0.0288 \text{ to } 0.0912 + 0.0288 \approx [0.0624, 0.1200] \]
This confidence interval provides a range in which we expect the true population proportion of inaccurate orders to lie, 95% of the time.