Question

1. a factory produces 2,000 red pencils every day. a sample of 60 pencils is analyzed, and 6 of them are found to not meet the standards of the factory, so they are labeled defective. a simulation is run in which 200 pencils are chosen out of 2,000 so that each pencil chosen has a 10% chance of being defective. the simulation is run 400 times, and the mean proportion of defective pencils is 0.098 with a standard deviation of 0.015. a. what is a good margin of error based on this simulation? b. based on the population proportion estimate and margin of error, is 0.13 a plausible value for the population proportion of pencils that are defective? explain your reasoning. 3. alex and jordan are studying the number of birds in a park. they each take a random sample to find the proportion of birds that are sparrows. alexs sample contains 15 birds, and jordans sample contains 25 birds. after collecting data, they run 100 simulations each to determine an estimate for the proportion of sparrows. how do you think alexs reported margin of error compares to jordans? explain your reasoning.

Explanation:

Step1: Recall margin - of - error formula

For a simulation, a common margin of error is \(3\times\) the standard deviation.

Step2: Calculate margin of error

Given standard deviation \(s = 0.015\), margin of error \(E=3\times0.015 = 0.045\).

Step3: Determine population proportion interval

The sample - based estimate of the population proportion is \(\hat{p}=0.098\). The confidence interval is \(\hat{p}\pm E\), so \(0.098 - 0.045=0.053\) and \(0.098 + 0.045 = 0.143\).

Step4: Analyze if 0.13 is plausible

Since \(0.053<0.13<0.143\), 0.13 is a plausible value for the population proportion.

Step5: Recall relationship between sample size and margin of error

The margin of error is inversely proportional to the square - root of the sample size (\(E\propto\frac{1}{\sqrt{n}}\)). Alex's sample size \(n_A = 15\) and Jordan's sample size \(n_J=25\). Since \(n_A < n_J\), Alex's margin of error will be larger.

Answer:

a. The margin of error is \(0.045\).
b. Yes, 0.13 is a plausible value for the population proportion of defective pencils because it lies within the confidence interval \((0.053, 0.143)\).

1. Alex's reported margin of error is larger than Jordan's because Alex's sample size (\(n = 15\)) is smaller than Jordan's sample size (\(n = 25\)) and margin of error is inversely proportional to the square - root of the sample size.