Problem 18 - Basic Statistics for Business and Economics with Student

Question

Schadek Silkscreen Printing, Inc., purchases plastic cups on which to print logos for sporting events, proms, birthdays, and other special occasions. Zack Schadek, the owner, received a large shipment this morning. To ensure the quality of the shipment, he selected a random sample of 300 cups. He found 15 to be defective. a. What is the estimated proportion defective in the population? b. Develop a 95 percent confidence interval for the proportion defective. c. Zack has an agreement with his supplier that he is to return lots that are 10 percent or more defective. Should he return this lot? Explain your decision.

Step-by-Step Solution

Answer

a. The estimated proportion defective is 0.05. b. 95% CI: (0.0253, 0.0747). c. Zack should not return the lot.

Step 1: Calculate the Sample Proportion

First, we need to compute the sample proportion defective \( \hat{p} \). According to the problem, Zack found 15 defective cups out of a sample of 300 cups. This gives us:\[ \hat{p} = \frac{15}{300} = 0.05. \]So, the estimated proportion of defective cups in the population is 0.05.

Step 2: Determine the Standard Error

The standard error (SE) of the sample proportion can be computed as:\[ SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}, \]where \( \hat{p} = 0.05 \) and \( n = 300 \). Substituting these values gives us:\[ SE = \sqrt{\frac{0.05 \times 0.95}{300}} = \sqrt{\frac{0.0475}{300}} \approx 0.0126. \]Thus, the standard error is approximately 0.0126.

Step 3: Calculate the Confidence Interval

To construct a 95% confidence interval for the proportion defective, use the formula:\[ CI = \hat{p} \pm Z \times SE, \]where \( Z \) is the critical value for a 95% confidence level, which is 1.96.So, the confidence interval is:\[ CI = 0.05 \pm 1.96 \times 0.0126. \]Calculating this gives us:\[ CI = (0.05 - 0.0247, 0.05 + 0.0247) = (0.0253, 0.0747). \]The 95% confidence interval for the proportion defective is from 0.0253 to 0.0747.

Step 4: Make a Decision About Returning the Lot

According to the 95% confidence interval, the upper limit is 0.0747, which is below 0.10. Therefore, we do not have enough evidence to suggest that the proportion of defective cups reaches 10% or more. Hence, Zack should not return the lot based on this evidence.

Key Concepts

Understanding Sample Proportion

The sample proportion is one of the foundational concepts in statistics and forms the basis for making estimations about a larger population. Imagine that instead of investigating every single item in a shipment, you take a smaller group, a sample, to examine. The sample proportion is the fraction of items in this sample that meet a certain criterion, like being defective. In Zack's case, he found 15 defective cups out of 300 total cups. This is calculated using the formula:

• \[ \hat{p} = \frac{\text{number of defective cups}}{\text{total number of cups}} = \frac{15}{300} = 0.05 \]

This result, \( \hat{p} = 0.05 \), tells us that 5% of the sampled cups were defective. The sample proportion provides an estimate, or best guess, of the proportion of defective items in the entire shipment. It is a critical step in determining whether differences are due to random sampling variation or indicate an actual issue with the entire batch.

Calculating the Standard Error

The standard error (SE) is an important measure that helps us understand how much variability there is in our sample proportion estimate. It tells us how much the sample proportion would fluctuate if we took many samples from the population. To compute the standard error of the sample proportion, the formula is:

• \[ SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \]

Where:- \( \hat{p} \) is the sample proportion (which is 0.05 in Zack's example),- \( n \) is the sample size (300 cups).
By substituting these values, the calculation becomes:

• \[ SE = \sqrt{\frac{0.05 \times 0.95}{300}} \approx 0.0126 \]

This standard error of approximately 0.0126 indicates the degree of sampling variability, effectively showing that different random samples from the population would produce proportions that are typically within about 0.0126 of 0.05.

The Role of Critical Value in Confidence Intervals

To develop a confidence interval, we need to apply the concept of a critical value. The critical value determines how much we should add and subtract from our sample proportion to create a range that we are a certain percentage confident contains the true population proportion. For instance, in a 95% confidence level, the critical value \( Z \) is 1.96, a standard value in statistics when the normal distribution is considered.

• Confidence Interval (CI) formula: \[ CI = \hat{p} \pm Z \times SE \]
• Plugging in Zack’s sample data:\[ CI = 0.05 \pm 1.96 \times 0.0126 = (0.0253, 0.0747) \]

The calculated confidence interval of (0.0253, 0.0747) means we are 95% confident that the true proportion of defective cups lies between 2.53% and 7.47%. Since 10% is not within this range, it suggests that the defect rate is likely below this threshold, guiding Zack's decision not to return the lot.