Question

1. You want to obtain a sample to estimate the proportion of a population that possess a particular genetic marker. Based on previous evidence, you believe approximately p∗=70%p∗=70% of the population have the genetic marker. You would like to be 98% confident that your estimate is within 5% of the true population proportion. How large of a sample size is required?

n =

2. Suppose we want to estimate the proportion of teenagers (aged 13-18) who are lactose intolerant. If we want to estimate this proportion to within 2% at the 95% confidence level, how many randomly selected teenagers must we survey?

3. You want to obtain a sample to estimate a population proportion. Based on previous evidence, you believe the population proportion is approximately p∗=28%p∗=28%. You would like to be 95% confident that your esimate is within 2% of the true population proportion. How large of a sample size is required?

n =

4. A researcher would like to estimate the proportion of children that have been diagnosed with a learning disability, such as dyslexia, dyscalculia or autism spectrum disorder (ASD) in their county. They randomly select 1000 children in their school districts and find that 10.6% have been diagnosed with a learning disability.

Construct a 90% confidence interval for the population proportion of all children that have been diagnosed with a learning disability in their county.

_____< __ <_____

Do not round between steps. Round answers to at least 4 decimal places

Answer 1

To estimate a population proportion, you can use the formula n = (Z^2 * p * (1-p)) / E^2, where Z is the Z-score corresponding to the desired confidence level, p is the estimated population proportion, and E is the margin of error. For each of the given questions, the required sample sizes can be calculated using this formula.

Step-by-step explanation:

To estimate a population proportion, the sample size required can be calculated using the formula n = (Z^2 * p * (1-p)) / E^2, where Z is the Z-score corresponding to the desired confidence level, p is the estimated population proportion, and E is the margin of error.

For question 1, if we want to be 98% confident that our estimate is within 5% of the true population proportion, the values would be Z = 2.33 (corresponding to 98% confidence), p = 0.7, and E = 0.05. Plugging these values into the formula, we get n = (2.33^2 * 0.7 * (1-0.7)) / 0.05^2 = 1485.68. Therefore, a sample size of at least 1486 would be required.

Similarly, for question 2, if we want to estimate the proportion of teenagers who are lactose intolerant to within 2% at the 95% confidence level, the values would be Z = 1.96 (corresponding to 95% confidence), p = unknown, and E = 0.02. Since we don't have an estimate for p, we can assume it to be 0.5 to get the largest required sample size. Plugging these values into the formula, we get n = (1.96^2 * 0.5 * (1-0.5)) / 0.02^2 = 9604. Therefore, a sample size of at least 9605 would be required.

For question 3, the values would be Z = 1.96 (corresponding to 95% confidence), p = 0.28, and E = 0.02. Plugging these values into the formula, we get n = (1.96^2 * 0.28 * (1-0.28)) / 0.02^2 = 619.8. Therefore, a sample size of at least 620 would be required.