a.
Construct a 90% confidence interval for the proportion of tenth-graders who plan to attend college using Wilson’s method.
Construct a 95% confidence interval for the proportion of tenth-graders who plan to attend college using Wilson’s method.
Construct a 99% confidence interval for the proportion of tenth-graders who plan to attend college using Wilson’s method.
a.
Answer to Problem 43E
The 90% confidence interval for the proportion of tenth-graders who plan to attend college using Wilson’s method is
The 95% confidence interval for the proportion of tenth-graders who plan to attend college using Wilson’s method is
The 99% confidence interval for the proportion of tenth-graders who plan to attend college using Wilson’s method is
Explanation of Solution
Calculation:
The given information is that,in a certain college 9 said that they planned to go to college after graduatingwhen 15 tenth-graders were asked.
Wilson’s interval:
For constructing a confidence interval the small-sample method is a simple approximation of very complicated interval that is, Wilson’s interval. Consider
Wilson’s confidence interval for p is given by,
Point estimate:
The point estimate
Substitute x as 9 and 15 as n in the formula,
Thus, the point estimate
For 90% confidence interval:
From the bottom row of Table A.3: Critical Values for the Student’s t Distribution, the critical value
Now, substitute
Thus, the 90% confidence interval for the proportion of tenth-graders who plan to attend college using Wilson’s method is
For 95% confidence interval:
From the bottom row of Table A.3: Critical Values for the Student’s t Distribution, the critical value
Now, substitute
Thus, the 95% confidence interval for the proportion of tenth-graders who plan to attend college using Wilson’s method is
For 99% confidence interval:
From the bottom row of Table A.3: Critical Values for the Student’s t Distribution, the critical value
Now, substitute
Thus, the 99% confidence interval for the proportion of tenth-graders who plan to attend college using Wilson’s method is
b.
Construct a 90% confidence interval for the proportion of tenth-graders who plan to attend college using small-sample method.
Construct a 95% confidence interval for the proportion of tenth-graders who plan to attend college using small-sample method.
Construct a 99% confidence interval for the proportion of tenth-graders who plan to attend college using small-sample method.
b.
Answer to Problem 43E
The 90% confidence interval for the proportion of tenth-graders who plan to attend college using small-sample method is
The 95% confidence interval for the proportion of tenth-graders who plan to attend college using small-sample method is
The 99% confidence interval for the proportion of tenth-graders who plan to attend college using small-sample method is
Explanation of Solution
Calculation:
Constructing confidence intervals for a proportion with small samples:
If x represents the number of individuals in a sample of size n that has certain characteristic and p is the population proportion, then
The adjusted sample proportion is,
The confidence interval for p is,
Substitute x as 9 and n as 15 in the formula of adjusted sample proportion,
For 90% confidence interval:
From the bottom row of Table A.3: Critical Values for the Student’s t Distribution, the critical value
Now, substitute
Thus, the 90% confidence interval for the proportion of tenth-graders who plan to attend college using small-sample method is
For 95% confidence interval:
From the bottom row of Table A.3: Critical Values for the Student’s t Distribution, the critical value
Now, substitute
Thus, the 95% confidence interval for the proportion of tenth-graders who plan to attend college using small-sample method is
For 99% confidence interval:
From the bottom row of Table A.3: Critical Values for the Student’s t Distribution, the critical value
Now, substitute
Thus, the 99% confidence interval for the proportion of tenth-graders who plan to attend college using small-sample method is
c.
Explain for which level the small-sample method is closer to Wilson’s method.
c.
Explanation of Solution
Approximation:
For Wilson’s method the small-sample method is a good approximation for all confidence levels commonly used in practice. And it is best when
From parts (a) and (b) it is observed that, the 95% confidence intervals obtained using Wilson’s method and small-sample method is same because the
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Chapter 7 Solutions
Essential Statistics
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