
Concept explainers
Expand Your knowledge: Alternate Method for Confidence Intervals When a is unknown and the sample is of size
Method 1: use the Student’s t distribution with d.f. = n - 1.
This is the method used in the text. It is widely employed in statistical studies Also, most statistical software packages use this method.
Mrthod 2: When
This method is based on the fact that for large samples, s is a fairly good approximation for
Consider a random sample of size n = 31, with sample mean x = 45.2 and sample standard deviation s = 5.3.
(a) Compute 90%, 95%, and 99% confidence intervals for µ using Method 1 with a Student’s t distribution. Round endpoints to two digits after the decimal.
(b) Compute 90%, 95% and 99% confidence intervals for µ using Method 2 with the standard normal distribute. Use s as an estimate for
(c) Compare intervals for the two methods. Would you say that confidence intervals using a Student’s t distribute are more conservative in the sense that they tend to be longer than intervals based on the standard normal distribute ?
(d) Repeal parts (a) through (c) for a sample of size n = 81. With increased
(a)

To find: The90%, 95%, and 99% confidence intervals for
Answer to Problem 23P
Solution:
The 90%, 95% and 99% confidence interval for
Explanation of Solution
Calculation:
Let
We have to find 90% confidence interval,
90% confidence interval is
The 90% confidence interval for
We have to find 95% confidence interval,
95% confidence interval is
The 95% confidence interval for
We have to find 99% confidence interval,
99% confidence interval is
The 99% confidence interval for
(b)

To find: The 90%, 95%, and 99% confidence intervals for
Answer to Problem 23P
Solution:
The 90%, 95% and 99% confidence interval for
Explanation of Solution
Calculation:
Let
We have to find 90% confidence interval,
90% confidence interval is
The 90% confidence interval for
We have to find 95% confidence interval,
95% confidence interval is
The 95% confidence interval for
We have to find 99% confidence interval,
99% confidence interval is
The 99% confidence interval for
(c)

Whether the confidenceintervals using a Student’s t distribution are more conservative.
Answer to Problem 23P
Solution:
Yes, the confidenceintervals using a Student’s t distribution are more conservative.
Explanation of Solution
Yes, the confidenceintervals using a Student’s t distribution is slightly longer than the intervals based on the standard normal distribution. Hence, the respective confidenceintervals using a Student’s t distribution are more conservative.
(d)

The 90%, 95%, and 99% confidence intervals for
Answer to Problem 23P
Solution:
The 90%, 95% and 99% confidence interval for
Explanation of Solution
Let
Using Method 1 with a Student’s t distribution:
We have to find 90% confidence interval,
90% confidence interval is
The 90% confidence interval for
We have to find 95% confidence interval,
95% confidence interval is
The 95% confidence interval for
We have to find 99% confidence interval,
99% confidence interval is
The 99% confidence interval for
Using Method 2 with the standard normal distribution:
We have to find 90% confidence interval,
90% confidence interval is
The 90% confidence interval for
We have to find 95% confidence interval,
95% confidence interval is
The 95% confidence interval for
We have to find 99% confidence interval,
99% confidence interval is
The 99% confidence interval for
The confidenceintervals using a Student’s t distribution is slightly longer than the intervals based on the standard normal distribution. As the sample size increases, the difference between the two methods is less pronounced. That means with increased sample size, the two methods give respective confidence intervals that are more similar.
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Chapter 8 Solutions
Understanding Basic Statistics
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