91Ó°ÊÓ

When we talk about a point estimate in statistics, we refer to a single value that serves as an estimate of a population parameter. In simpler terms, it is a best guess we can derive from the sample data about the characteristic of a larger group.
For the exercise in question, the point estimate is the proportion of packages weighing at least 490 grams from our sample data. Given that 288 out of 300 packages weighed as much, our point estimate

• is calculated by dividing the favorable cases (packages meeting the weight requirement) by the total number of samples, resulting in \( \hat{p} = \frac{288}{300} \).
• This calculation gives \( \hat{p} = 0.96 \), meaning 96% of the sampled packages meet the weight condition.

This point estimate provides us a snapshot of what we can expect in the whole population based on our sample.

Before constructing a confidence interval, it's crucial to ensure that the sample size is adequate. This is a fundamental concept in statistics because it affects the reliability and validity of our estimates.
In our example, the conditions for sample size sufficiency are:

• First, we calculate \( n\hat{p} \) as \( 300 \times 0.96 = 288 \).
• Next, we find \( n(1-\hat{p}) \) as \( 300 \times 0.04 = 12 \).

Both calculations ensure that the criteria \( n\hat{p} \geq 5 \) and \( n(1-\hat{p}) \geq 5 \) are satisfied, which confirms our sample is large enough.
If these conditions were not met, our confidence interval could be unreliable.

The critical value is a crucial component when calculating a confidence interval. It tells us the number of standard deviations we need to go from the mean to capture a desired percentage of the data in a normal distribution.
For constructing a confidence interval, especially at a high confidence level like 99.8%, the critical value must be carefully selected using a z-table or standard normal distribution table.

• In our case, for a 99.8% confidence interval, the critical value (\( z \)) is approximately 3.090.
• The critical value allows us to determine the margin of error in our confidence interval calculation.

We then use the formula:\[CI = \hat{p} \pm z \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\]Inserting the values from the exercise gives a confidence interval of \((0.941, 0.979)\), which interprets to having a high degree of certainty that the actual proportion of all packages meeting the weight requirement falls within this range. This application of critical value helps to express the reliability and precision of our estimations.