When σ is unknown and the sample is of size
n
≥ 30, there are two methods for computing confidence intervals for μ.
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.
Method 2: When
n
≥ 30, use the sample standard deviation
s
as an estimate for σ, and then use the standard normal distribution.
This method is based on the fact that for large samples,
s
is a fairly good approximation for σ. Also, for large
n, the critical values for the Student's
t
distribution approach those of the standard normal distribution.
Consider a random sample of size
n
= 36, with sample mean x = 45.6 and sample standard deviation
s
= 6.5.
(d) Now consider a sample size of 81. 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.
|
90% |
95% |
99% |
lower limit |
|
|
|
upper limit |
|
|
|
(e) Compute 90%, 95%, and 99% confidence intervals for μ using Method 2 with the standard normal distribution. Use
s
as an estimate for σ. Round endpoints to two digits after the decimal.
|
90% |
95% |
99% |
lower limit |
|
|
|
upper limit |
|
|
|