Concept explainers
Wilson’s interval: The small-sample method for constructing a confidence interval is a simple, approximation of a more complicated interval known as Wilson’s interval. Let
Wilson’s confidence interval for p is given by
Approximation depends on the level: The small-sample method is a good approximation to Wilson’s method for all confidence levels commonly used in practice, but is best when is close to 2. Refer to Exercise 41.
- Use Wilson’s method to construct a 90% confidence interval, a 95% confidence interval, and a 99% confidence Interval for the proportion of tenth graders who plan to attend college.
- Use the small-sample method to construct a 90% confidence interval, a 95% confidence interval, and a 99% confidence interval for the proportion of tenth graders who plan to attend college.
- For which level is the small-sample method the closest to Wilson’s method? Explain why this is the case.
(a)
Use Wilson’s method to construct the confidence interval.
Answer to Problem 43E
At 90% confidence interval the probability value is
At 95% confidence interval the probability value is
At 99% confidence interval the probability value is
Explanation of Solution
The Wilson’s method formula is given,
Here, p is the probability value and z are critical value
Then compute the Wilson’s confidence interval method
At 90% confidence interval the probability value is
At 95% confidence interval the probability value is
At 99% confidence interval the probability value is
Program:
clc clear close all n=15; x=9; p=9/15; z1=1.645; z2=1.96; z3=2.575; up90=p+(z1/(2*n))+z1*sqrt((p*(1-p)/n)+((z1)^2/(4*n^2))); up95=p+(z2/(2*n))+z2*sqrt((p*(1-p)/n)+((z2)^2/(4*n^2))); up99=p+(z3/(2*n))+z3*sqrt((p*(1-p)/n)+((z3)^2/(4*n^2))); div90=1+(z1^2/n); div95=1+(z2^2/n); div99=1+(z3^2/n); w90p=up90/div90 w95p=up95/div95 w99p=up99/div99 low90=p+(z1/(2*n))-z1*sqrt((p*(1-p)/n)+((z1)^2/(4*n^2))); low95=p+(z2/(2*n))-z2*sqrt((p*(1-p)/n)+((z2)^2/(4*n^2))); low99=p+(z3/(2*n))-z3*sqrt((p*(1-p)/n)+((z3)^2/(4*n^2))); w90n=low90/div90 w95n=low95/div95 w99n=low99/div99
(b)
Using small-sample method to construct the confidence interval.
Answer to Problem 43E
At 90% confidence interval the probability value is
At 95% confidence interval the probability value is
At 99% confidence interval the probability value is
Explanation of Solution
The small-sample method formula is given,
Here, p is the probability value and z are critical value
Then compute the Wilson’s confidence interval method
At 90% confidence interval the probability value is
At 95% confidence interval the probability value is
At 99% confidence interval the probability value is
Program:
clc clear close all n=15; x=9; p=9/15; z1=1.645; z2=1.96; z3=2.575; up90=p+z1*sqrt((p*(1-p)/(n+4))) up95=p+z2*sqrt((p*(1-p)/(n+4))) up99=p+z3*sqrt((p*(1-p)/(n+4))) low90=p-z1*sqrt((p*(1-p)/(n+4))) low95=p-z2*sqrt((p*(1-p)/(n+4))) low99=p-z3*sqrt((p*(1-p)/(n+4)))
(c)
Explain at which interval Wilson’s method close to small-sample method.
Answer to Problem 43E
At the 90 % interval Wilson’s method close to small-sample method.
The Wilson’s method shows
At 90% confidence interval the probability value is
The Small-sample method shows
At 90% confidence interval the probability value is
Explanation of Solution
Wilson’s method formula
The small-sample formula
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Chapter 8 Solutions
Elementary Statistics ( 3rd International Edition ) Isbn:9781260092561
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