Question

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 = 45.1 and sample standard deviation s = 4.5.(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?

Yes. The respective intervals based on the t distribution are shorter.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.

(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

(f) 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?

Yes. The respective intervals based on the t distribution are shorter.Yes. The respective intervals based on the t distribution are longer.    No. The respective intervals based on the t distribution are longer.No. The respective intervals based on the t distribution are shorter.

With increased sample size, do the two methods give respective confidence intervals that are more similar?

As the sample size increases, the difference between the two methods is less pronounced.As the sample size increases, the difference between the two methods remains constant.    As the sample size increases, the difference between the two methods becomes greater.

Answer 1

a) df = 36 - 1 = 35

At 90% confidence interval the critical value is t^* = 1.69

The 90% confidence interval is

+/- t^* * s/

= 45.1 +/- 1.69 * 4.5/

= 45.1 +/- 1.27

= 43.83, 46.37

Lower limit = 43.83

Upper limit = 46.37

At 95% confidence interval the critical value is t^* = 2.03

The 95% confidence interval is

+/- t^* * s/

= 45.1 +/- 2.03 * 4.5/

= 45.1 +/- 1.52

= 43.58, 46.62

Lower limit = 43.58

Upper limit = 46.62

At 99% confidence interval the critical value is t^* = 2.724

The 99% confidence interval is

+/- t^* * s/

= 45.1 +/- 2.724 * 4.5/

= 45.1 +/- 2.04

= 43.06, 47.14

Lower limit = 43.06

Upper limit = 47.14

b) At 90% confidence interval the critical value is z^* = 1.645

The 90% confidence interval is

+/- z^* * s/

= 45.1 +/- 1.645 * 4.5/

= 45.1 +/- 1.23

= 43.87, 46.33

Lower limit = 43.87

Upper limit = 46.33

At 95% confidence interval the critical value is z^* = 1.96

The 95% confidence interval is

+/- z^* * s/

= 45.1 +/- 1.96 * 4.5/

= 45.1 +/- 1.47

= 43.63, 46.57

Lower limit = 43.63

Upper limit = 46.57

At 99% confidence interval the critical value is z^* = 2.575

The 99% confidence interval is

+/- z^* * s/

= 45.1 +/- 2.575 * 4.5/

= 45.1 +/- 1.93

= 43.17, 47.03

Lower limit = 43.17

Upper limit = 47.03

c) df = 81 - 1 = 80

At 90% confidence interval the critical value is t^* = 1.664

The 90% confidence interval is

+/- t^* * s/

= 45.1 +/- 1.664 * 4.5/

= 45.1 +/- 0.83

= 44.27, 45.93

Lower limit = 44.27

Upper limit = 45.93

At 95% confidence interval the critical value is t^* = 1.99

The 95% confidence interval is

+/- t^* * s/

= 45.1 +/- 1.99 * 4.5/

= 45.1 +/- 0.995

= 44.105, 46.095

= 44.11, 46.10

Lower limit = 44.11

Upper limit = 46.10

At 99% confidence interval the critical value is t^* = 2.639

The 99% confidence interval is

+/- t^* * s/

= 45.1 +/- 2.639 * 4.5/

= 45.1 +/- 1.32

= 43.78, 46.42

Lower limit = 43.78

Upper limit = 46.42

d) At 90% confidence interval the critical value is z^* = 1.645

The 90% confidence interval is

+/- z^* * s/

= 45.1 +/- 1.645 * 4.5/

= 45.1 +/- 0.82

= 44.28, 45.92

Lower limit = 44.28

Upper limit = 45.92

At 95% confidence interval the critical value is z^* = 1.96

The 95% confidence interval is

+/- z^* * s/

= 45.1 +/- 1.96 * 4.5/

= 45.1 +/- 0.98

= 44.12, 46.08

Lower limit = 44.12

Upper limit = 46.08

At 99% confidence interval the critical value is z^* = 2.575

The 99% confidence interval is

+/- z^* * s/

= 45.1 +/- 2.575 * 4.5/

= 45.1 +/- 1.29

= 43.81, 46.39

Lower limit = 43.81

Upper limit = 46.39

f) No. The respective intervals based on the t distribution are longer.

As the sample size increases the difference between the two methods is less pronounced.