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 = 45.7 and sample standard deviation s = 6.0.

(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) 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

(d) 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

Answer 1

a) At 90% confidence level the critical value is t^* = 1.69

The 90% confidence interval is

+/- t^* * s/

= 45.7 +/- 1.69 * 6/

= 45.7 +/- 1.69

= 44.01, 47.39

Lower limit = 44.01

Upper limit = 47.39

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

The 95% confidence interval is

+/- t^* * s/

= 45.7 +/- 2.03 * 6/

= 45.7 +/- 2.03

= 43.67, 47.73

Lower limit = 43.67

Upper limit = 47.73

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

The 99% confidence interval is

+/- t^* * s/

= 45.7 +/- 2.724 * 6/

= 45.7 +/- 2.72

= 42.98, 48.42

Lower limit = 42.98

Upper limit = 48.42

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

The 90% confidence interval is

+/- z^* * s/

= 45.7 +/- 1.645 * 6/

= 45.7 +/- 1.645

= 44.055, 47.345

= 44.06, 47.35

Lower limit = 44.06

Upper limit = 47.35

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

The 95% confidence interval is

+/- z^* * s/

= 45.7 +/- 1.96 * 6/

= 45.7 +/- 1.96

= 43.74, 47.66

Lower limit = 43.74

Upper limit = 47.66

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

The 99% confidence interval is

+/- z^* * s/

= 45.7 +/- 2.575 * 6/

= 45.7 +/- 2.575

= 43.125, 48.275

= 43.13, 48.28

Lower limit = 43.13

Upper limit = 48.28

c) At 90% confidence level the critical value is t^* = 1.667

The 90% confidence interval is

+/- t^* * s/

= 45.7 +/- 1.667 * 6/

= 45.7 +/- 1.19

= 44.51, 46.89

Lower limit = 44.51

Upper limit = 46.89

At 95% confidence level the critical value is t^* = 1.994

The 95% confidence interval is

+/- t^* * s/

= 45.7 +/- 1.994 * 6/

= 45.7 +/- 1.42

= 44.28, 47.12

Lower limit = 44.28

Upper limit = 47.12

At 99% confidence level the critical value is t^* = 2.648

The 99% confidence interval is

+/- t^* * s/

= 45.7 +/- 2.648 * 6/

= 45.7 +/- 1.89

= 43.81, 47.59

Lower limit = 43.81

Upper limit = 47.59

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

The 90% confidence interval is

+/- z^* * s/

= 45.7 +/- 1.645 * 6/

= 45.7 +/- 1.17

= 44.53, 46.87

Lower limit = 44.53

Upper limit = 46.87

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

The 95% confidence interval is

+/- z^* * s/

= 45.7 +/- 1.96 * 6/

= 45.7 +/- 1.40

= 43.30, 47.10

Lower limit = 43.30

Upper limit = 47.10

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

The 99% confidence interval is

+/- z^* * s/

= 45.7 +/- 2.575 * 6/

= 45.7 +/- 1.83

= 43.87, 47.53

Lower limit = 43.87

Upper limit = 47.53