91Ó°ÊÓ

In the world of statistics, the sample proportion is a crucial concept, especially when estimating characteristics of a population based on a sample. The sample proportion, denoted as \( \hat{p} \), is essentially the ratio of items in the sample that belong to a particular category of interest. In our exercise, this category is the defective plastic cups. If Zack collected a sample of 300 cups and found 15 were defective, the calculation of the sample proportion becomes straightforward:

• Number of defective cups = 15
• Total number of cups in the sample = 300

The sample proportion is calculated as follows: \[\hat{p} = \frac{15}{300} = 0.05\] This result tells us that approximately 5% of the cups in Zack's sample are defective.
This estimated proportion helps us infer about the level of defects in the entire shipment and aids in further statistical analysis.

The standard error of a statistic gives us a measure of how much that statistic is expected to vary from sample to sample, due to sampling variability. For a sample proportion, the standard error (SE) can be calculated using the formula: \[ SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \]where \( n \) is the sample size.
Using Zack's data:

• Sample proportion \( \hat{p} = 0.05 \)
• Sample size \( n = 300 \)

Plug these values into the formula:\[SE = \sqrt{\frac{0.05 \times (1 - 0.05)}{300}} \approx 0.0126\]This number, 0.0126, indicates the expected variability in the sample proportion from different samples of the same size from the population.
The smaller the standard error, the more precise our estimate of the population proportion is.

Defective proportion refers to the part of the shipment that does not meet the required quality standards. In Zack's context, it's the proportion of plastic cups that are faulty or unsatisfactory. Calculating and understanding this proportion is vital for making data-driven business decisions.
Zack's goal is to use the estimated defective proportion from his sample to make inferences about the entire shipment.
By calculating a 95% confidence interval for the defective proportion, Zack can estimate a range where the true defective proportion likely lies:

• Estimated defective proportion (\( \hat{p} \)) = 0.05
• Confidence interval calculated to be [0.0253, 0.0747]

This confidence interval suggests that, although 5% of the sample was defective, the actual proportion in the entire shipment could reasonably be anywhere from 2.53% to 7.47%.
As this is below 10%, Zack concludes that returning the shipment might not be statistically justified.

The Z-distribution is a type of statistical distribution that follows the standard normal distribution, ideal for handling population statistics when the sample size is large. It allows statisticians to determine probabilities and critical values used in hypothesis testing and confidence interval construction.
In our exercise, the Z-distribution helps in constructing a confidence interval for Zack's defective cup proportion. When creating a confidence interval estimate for a proportion, the critical value from the Z-distribution is used:

• The critical value for a 95% confidence level is 1.96.

This value reflects the range within which we are 95% confident that the true population proportion lies.
Zack used it to compute the limits of his confidence interval:\[ p \pm 1.96 \times SE \]Substituting his sample proportion and standard error:

• Lower bound: 0.05 - 1.96 × 0.0126 = 0.0253
• Upper bound: 0.05 + 1.96 × 0.0126 = 0.0747

This enables Zack to confidently gauge the proportion of defects, ensuring quality standards are efficiently checked and maintained.