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Solved: To deal with issues such as the distribution of p
Chapter 5, Problem 44AYU(choose chapter or problem)
Problem 44AYU
To deal with issues such as the distribution of p not following a normal distribution A. Agresti and B. Coull (Approximate Is Better Than “Exact” for Interval Estimation of Binomial Proportion. American Statistician, 52:119-26, 1998) proposed a modified approach to constructing confidence intervals for a proportion. A (1 – α) • 100% confidence interval for p is given by
Lower bound:
Upper bound:
where (x is the number of successes in n trials).
Use this result to answer Problem 1-2.
Problem 1
Walk to Work Alan wants to estimate the proportion of adults who walk to work. In a survey of 10 adults, he finds 0 who walk to work. Explain why a 95% confidence interval using the normal model yields silly results. Then compute and interpret a 95% confidence interval for the proportion of adults who walk to work using Agresti and Coull’s method.
Problem 2
Simulation - When Model Requirements Fail A Bernoulli random variable is a variable that is either 0 (a failure) or 1 (a success). The probability of success is denoted p.
(a) Use StatCrunch, MINITAB, or some other statistical spreadsheet to generate 1000 Bernoulli samples of size n = 20 with p = 0.15.
(b) Estimate the population proportion for each of the 1000 Bernoulli samples.
(c) Draw a histogram of the 1000 proportions from part (b). What is the shape of the histogram?
(d) Construct a 95% confidence interval for each of the 1000 Bernoulli samples using the normal model.
(e) What proportion of the intervals do you expect to include the population proportion, p? What proportion of the intervals actually captures the population proportion? Explain any differences.
Questions & Answers
QUESTION:
Problem 44AYU
To deal with issues such as the distribution of p not following a normal distribution A. Agresti and B. Coull (Approximate Is Better Than “Exact” for Interval Estimation of Binomial Proportion. American Statistician, 52:119-26, 1998) proposed a modified approach to constructing confidence intervals for a proportion. A (1 – α) • 100% confidence interval for p is given by
Lower bound:
Upper bound:
where (x is the number of successes in n trials).
Use this result to answer Problem 1-2.
Problem 1
Walk to Work Alan wants to estimate the proportion of adults who walk to work. In a survey of 10 adults, he finds 0 who walk to work. Explain why a 95% confidence interval using the normal model yields silly results. Then compute and interpret a 95% confidence interval for the proportion of adults who walk to work using Agresti and Coull’s method.
Problem 2
Simulation - When Model Requirements Fail A Bernoulli random variable is a variable that is either 0 (a failure) or 1 (a success). The probability of success is denoted p.
(a) Use StatCrunch, MINITAB, or some other statistical spreadsheet to generate 1000 Bernoulli samples of size n = 20 with p = 0.15.
(b) Estimate the population proportion for each of the 1000 Bernoulli samples.
(c) Draw a histogram of the 1000 proportions from part (b). What is the shape of the histogram?
(d) Construct a 95% confidence interval for each of the 1000 Bernoulli samples using the normal model.
(e) What proportion of the intervals do you expect to include the population proportion, p? What proportion of the intervals actually captures the population proportion? Explain any differences.
ANSWER:
Answer :
Step 1:
For the given information, Alan wants to estimate the proportion of adults who walk to work. In a survey of 10 adults, he finds 0 who walk to work.
Here given values are x = 0 and n = 10.
For the normal model sample proportion is 0, the upper and lower bounds of the confidence interval using the normal model are both zero.
Using Agresti and Coull’s method,
The formula for confidence interval is
Here = 0 and n = 10 , z* = 1.96 (from standard normal table)
lower bound: -0.040, upper bound: 0.326. We are 95% confident that the proportion of adults who walk to work is between 0 and 0.326.