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 = 41, with sample mean x = 45.7 and sample standard deviation s = 6.4.

(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

Method 1: Use the Student's t distribution with d.f. = n − 1.

Consider a random sample of size n = 41, with sample mean x = 45.7 and sample standard deviation s = 6.4.

Formula for Confidence Interval

90% confidence intervals for μ using Method 1 with a Student's t distribution

Sample Size : n	41
Degrees of freedom : n-1	40
Confidence Level :	90
	0.1
/2	0.05
	1.6839

95% confidence intervals for μ using Method 1 with a Student's t distribution

Sample Size : n	41
Degrees of freedom : n-1	40
Confidence Level :	95
	0.05
/2	0.025
t_{0.025,40}	2.0211

99% confidence intervals for μ using Method 1 with a Student's t distribution

Degrees of freedom : n-1	40
Confidence Level :	99
	0.01
/2	0.005
t_{0.005,40}	2.7045

	90%	95%	99%
Lower Limit	44.02	43.68	43
Upper Limit	47.38	47.72	48.4

(b) Compute 90%, 95%, and 99% confidence intervals for μ using Method 2 with the standard normal distribution

Formula

90% confidence intervals for μ using Method 2 with the standard normal distribution

Confidence Level :	90
:	0.1
/2	0.05
Z0.05	1.6449

95% confidence intervals for μ using Method 2 with the standard normal distribution

Confidence Level :	95
	0.05
/2	0.025
Z0.025	1.96

99% confidence intervals for μ using Method 2 with the standard normal distribution

Confidence Level :	99
	0.01
/2	0.005
Z0.005	2.5758

	90%	95%	99%
Lower Limit	44.06	43.74	43.13
Upper Limit	47.34	47.66	48.27

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

Formula for Confidence Interval

90% confidence intervals for μ using Method 1 with a Student's t distribution

Sample Size : n	71
Degrees of freedom : n-1	70
Confidence Level :	90
	0.1
/2	0.05
t0.05,70	1.6669

95% confidence intervals for μ using Method 1 with a Student's t distribution

Sample Size : n	71
Degrees of freedom : n-1	70
Confidence Level :	95
	0.05
/2	0.025
t0.025,70	1.9944

99% confidence intervals for μ using Method 1 with a Student's t distribution

Sample Size : n	71
Degrees of freedom : n-1	70
Confidence Level :	99
	0.01
/2	0.005
t0.005,70	2.6479

	90%	95%	99%
Lower Limit	44.43	44.19	43.69
Upper Limit	46.97	47.21	47.71

(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

Formula

90% confidence intervals for μ using Method 2 with the standard normal distribution

Confidence Level :	90
:	0.1
/2	0.05
Z0.05	1.6449

95% confidence intervals for μ using Method 2 with the standard normal distribution

Confidence Level :	95
	0.05
/2	0.025
Z0.025	1.96

99% confidence intervals for μ using Method 2 with the standard normal distribution

Confidence Level :	99
	0.01
/2	0.005
Z0.005	2.5758

	90%	95%	99%
Lower Limit	44.45	44.21	43.74
Upper Limit	46.95	47.19	47.66