A company’s CEO wanted to estimate the percentage of defective product per shipment. In a sample containing 500 products, he found 30 defective products.
(a) Find a 98% confidence interval for the true proportion of defective product. Show your calculations and/or explain the process used to obtain the interval.
(b) Interpret this confidence interval and write a sentence that explains it.
To find a 98% confidence interval for the true proportion of defective products per shipment, we can use the formula for a confidence interval for a proportion:
Confidence interval = sample proportion ± (critical value) x (standard error)
(a) Calculating the confidence interval:
First, we need to calculate the sample proportion, which is the number of defective products divided by the total sample size:
Sample proportion (p-hat) = Number of defective products / Total sample size
= 30 / 500
= 0.06
Next, we need to determine the critical value for a 98% confidence level. This value can be found in a standard normal distribution table or using statistical software. For a two-tailed test at a 98% confidence level, the critical value is approximately 2.33.
The standard error, which measures the variability in the sample proportion, can be calculated using the formula:
Standard error = √(p-hat * (1 - p-hat) / n)
= √(0.06 * (1 - 0.06) / 500)
= √(0.0576 / 500)
= √0.0001152
= 0.0107 (rounded to four decimal places)
Now we can calculate the confidence interval:
Confidence interval = 0.06 ± (2.33 * 0.0107)
= 0.06 ± 0.0249
= (0.0351, 0.0849)
Therefore, the 98% confidence interval for the true proportion of defective products per shipment is (0.0351, 0.0849).
(b) Interpretation of the confidence interval:
We are 98% confident that the true proportion of defective products per shipment lies within the range of 0.0351 to 0.0849. This means that if we were to take multiple samples and construct 98% confidence intervals for each sample, 98% of those intervals would contain the true proportion of defective products. In practical terms, the interpretation suggests that the CEO can be quite confident that the percentage of defective products in the population falls within this range.