Question
In: Statistics and Probability
When σ is unknown and the sample is of size n ≥ 30, there are two...
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 = 45.8 and sample standard deviation s = 6.3.
(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.
| 90% | 95% | 99% | |
| lower limit | |||
| upper limit |
(b) 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 |
(c) Compare intervals for the two methods. Would you say that
confidence intervals using a Student's t distribution are
more conservative in the sense that they tend to be longer than
intervals based on the standard normal distribution?
No. The respective intervals based on the t distribution are longer.Yes. The respective intervals based on the t distribution are longer. No. The respective intervals based on the t distribution are shorter.Yes. The respective intervals based on the t distribution are shorter.
(d) Now consider a sample size of 71. 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 |
Solutions
orchestra answered 2 years agoRelated Solutions
When σ is unknown and the sample is of size n ≥ 30, there are two...
When σ is unknown and the sample is of size n ≥ 30, there are two...
When σ is unknown and the sample is of size n ≥ 30, there are two...
When σ is unknown and the sample is of size n ≥ 30, there are two...
When σ is unknown and the sample is of size n ≥ 30, there are two...
When σ is unknown and the sample is of size n ≥ 30, there are two...
When σ is unknown and the sample is of size n ≥ 30, there are two...
When σ is unknown and the sample is of size n ≥ 30, there are two...
When σ is unknown and the sample is of size n ≥ 30, there are two...
When σ is unknown and the sample is of size n ≥ 30, there are two...
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