Lock5stat

by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed

under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18

189

Problem Set Section 2E Directions: Answer the following questions.

1. What are the assumptions necessary for making a one-population proportion confidence interval?

2. What are the assumptions necessary for making a one-population mean confidence interval?

3. What are the assumptions necessary for making a one-population bootstrap confidence interval?

4. An experiment was conducted to see what percentage of rats would show empathy toward fellow rats in distress. Of the 30 total rats in the study, 23 showed empathy. What was the sample proportion? What are the critical value Z-scores for 99% confidence? If you cannot remember them, open StatKey at www.lock5stat.com. Go to “theoretical distributions” and click on “normal”. You can look up the critical value Z-scores. Use the critical values and the given standard error to calculate the margin of error and construct a 99% confidence interval estimate of the population proportion of rats that show empathy. Convert the upper and lower limits of your confidence interval into percentages.

Standard Error ≈ 0.07725

a) Sample Proportion �̂�𝑝 = 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝑜𝑜𝑜𝑜 𝑆𝑆𝑁𝑁𝑆𝑆𝑆𝑆𝑁𝑁𝑆𝑆𝑆𝑆 (𝑁𝑁𝑒𝑒𝑁𝑁𝑒𝑒𝑒𝑒𝑆𝑆) 𝑇𝑇𝑜𝑜𝑒𝑒𝑇𝑇𝑇𝑇 𝑆𝑆𝑇𝑇𝑁𝑁𝑆𝑆𝑇𝑇𝑁𝑁 𝑆𝑆𝑆𝑆𝑆𝑆𝑁𝑁

=

b) Critical value Z-scores = ±

c) Margin of Error = 𝑍𝑍× 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 𝐸𝐸𝑆𝑆𝑆𝑆𝐸𝐸𝑆𝑆 =

d) Confidence Interval Lower Limit = �̂�𝑝 − (𝑀𝑀𝑆𝑆𝑆𝑆𝑀𝑀𝑀𝑀𝑆𝑆 𝐸𝐸𝑜𝑜 𝐸𝐸𝑆𝑆𝑆𝑆𝐸𝐸𝑆𝑆)

e) Confidence Interval Upper Limit = �̂�𝑝 + (𝑀𝑀𝑆𝑆𝑆𝑆𝑀𝑀𝑀𝑀𝑆𝑆 𝐸𝐸𝑜𝑜 𝐸𝐸𝑆𝑆𝑆𝑆𝐸𝐸𝑆𝑆)

5. Use the following Statcato printout to check your margin of error and confidence interval answers from the rat empathy data in number 4. Now check the assumptions and write sentences to explain the margin of error and confidence interval.

a) Check each of the assumptions for this problem. Assume the rats were randomly selected. Explain your answers.

b) Write a sentence to explain the margin of error in context.

c) Write a sentence to explain the confidence interval in context.

https://creativecommons.org/licenses/by/4.0/
http://www.lock5stat.com/
This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed

under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18

190

6. A study was done on the effectiveness of l ie detector tests to catch someone that l ies. In a random sample of 48 total l ies, the machine identified only 31 of them. What was the sample proportion? What are the critical value Z-scores for 95% confidence? If you cannot remember them, open StatKey at www.lock5stat.com. Go to “theoretical distributions” and click on “normal”. You can look up the critical value Z-scores. Use the critical values and the given standard error to calculate the margin of error and construct a 95% confidence interval estimate of the population proportion of l ies caught be l ie detector tests. Convert the upper and lower l imits of your confidence interval into percentages.

Standard Error ≈ 0.0689

a) Sample Proportion �̂�𝑝 = 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝑜𝑜𝑜𝑜 𝑆𝑆𝑁𝑁𝑆𝑆𝑆𝑆𝑁𝑁𝑆𝑆𝑆𝑆 (𝑁𝑁𝑒𝑒𝑁𝑁𝑒𝑒𝑒𝑒𝑆𝑆) 𝑇𝑇𝑜𝑜𝑒𝑒𝑇𝑇𝑇𝑇 𝑆𝑆𝑇𝑇𝑁𝑁𝑆𝑆𝑇𝑇𝑁𝑁 𝑆𝑆𝑆𝑆𝑆𝑆𝑁𝑁

=

b) Critical value Z-scores = ±

c) Margin of Error = 𝑍𝑍× 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 𝐸𝐸𝑆𝑆𝑆𝑆𝐸𝐸𝑆𝑆 =

d) Confidence Interval Lower Limit = �̂�𝑝 − (𝑀𝑀𝑆𝑆𝑆𝑆𝑀𝑀𝑀𝑀𝑆𝑆 𝐸𝐸𝑜𝑜 𝐸𝐸𝑆𝑆𝑆𝑆𝐸𝐸𝑆𝑆)

e) Confidence Interval Upper Limit = �̂�𝑝 + (𝑀𝑀𝑆𝑆𝑆𝑆𝑀𝑀𝑀𝑀𝑆𝑆 𝐸𝐸𝑜𝑜 𝐸𝐸𝑆𝑆𝑆𝑆𝐸𝐸𝑆𝑆)

7. Use the following Statcato printout to check your margin of error and confidence interval answers from the l ie detector data in number 6. Now check the assumptions and write sentences to explain the margin of error and confidence interval.

a) Check each of the assumptions for this problem. Explain your answers.

b) Write a sentence to explain the margin of error in context.

c) Write a sentence to explain the confidence interval in context.

https://creativecommons.org/licenses/by/4.0/
http://www.lock5stat.com/
This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed

under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18

191

8. We want to determine what percentage of cereals the company Quaker makes. A random sample of 24 cereals found that Quaker made four of them. What was the sample proportion? What are the critical value Z-scores for 90% confidence? If you cannot remember them, open StatKey at www.lock5stat.com. Go to “theoretical distributions” and click on “normal”. You can look up the critical value Z-scores. Use the critical values and the given standard error to calculate the margin of error and construct a 90% confidence interval estimate of the population proportion of cereals made by Quaker. Convert the upper and lower limits of your confidence interval into percentages.

Standard Error ≈ 0.076

a) Sample Proportion �̂�𝑝 = 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝑜𝑜𝑜𝑜 𝑆𝑆𝑁𝑁𝑆𝑆𝑆𝑆𝑁𝑁𝑆𝑆𝑆𝑆 (𝑁𝑁𝑒𝑒𝑁𝑁𝑒𝑒𝑒𝑒𝑆𝑆) 𝑇𝑇𝑜𝑜𝑒𝑒𝑇𝑇𝑇𝑇 𝑆𝑆𝑇𝑇𝑁𝑁𝑆𝑆𝑇𝑇𝑁𝑁 𝑆𝑆𝑆𝑆𝑆𝑆𝑁𝑁

=

b) Critical value Z-scores = ±

c) Margin of Error = 𝑍𝑍× 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 𝐸𝐸𝑆𝑆𝑆𝑆𝐸𝐸𝑆𝑆 =

d) Confidence Interval Lower Limit = �̂�𝑝 − (𝑀𝑀𝑆𝑆𝑆𝑆𝑀𝑀𝑀𝑀𝑆𝑆 𝐸𝐸𝑜𝑜 𝐸𝐸𝑆𝑆𝑆