A population consists of a batch of 25,664 aspirin tablets, and it includes 994 that are defective because they do not meet specifications. A random sample of \( n=220 \) of the tablets is obtained and tested, with the result that 15 of them are defective. a. What is the population proportion, \( p \), of defective aspirin tablets? b. What is the sample proportion, \( \hat{p} \), of defective tablets? c. How can the discrepancy between the population proportion and the sample proportion be explained? a. The population proportion of defective aspirin tablets is \( p=\square \). (Round to three decimal places as needed.)

To find the population proportion of defective aspirin tablets, we use the formula: \[ p = \frac{\text{number of defective tablets}}{\text{total number of tablets}} = \frac{994}{25,664} \] Calculating this gives: \[ p \approx 0.0388 \] Rounded to three decimal places, the population proportion of defective aspirin tablets is \( p = 0.039 \). For the sample proportion \( \hat{p} \): \[ \hat{p} = \frac{\text{number of defective tablets in the sample}}{\text{total number of tablets in the sample}} = \frac{15}{220} \] Calculating this gives: \[ \hat{p} \approx 0.0682 \] Rounded to three decimal places, the sample proportion of defective tablets is \( \hat{p} = 0.068 \). The discrepancy between the population proportion and the sample proportion can be explained by random sampling variability. Samples can sometimes over-represent or under-represent certain characteristics of a population purely by chance. In this case, the sample contained a higher proportion of defective tablets than what is present in the overall population, which is a common occurrence in statistical sampling.