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 = 31, with sample mean x = 44.3 and sample standard deviation s = 6.5. (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.(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.(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.

Answer 1

a)

Degree of freedom: df=n-1=30

Excel function used for critical value of t: "=ROUND(TINV(1-C,df),3)"

Here C is confidence level. For example for 90% confidence interval C= 0.90.

Following table shows the critical values of t :

		Critical value
df	90%	95%	99%
30	1.697	2.042	2.75

The formula for confidence interval is

Following table shows the requried confidence interval:

	90%	95%	99%
Lower limit	42.32	41.92	41.09
Upper limit	46.28	46.68	47.51

(b)

Following table shows the critical values of z :

	Critical value
90%	95%	99%
1.645	1.96	2.575

The formula for confidence interval is

Following table shows the requried confidence interval:

	90%	95%	99%
Lower limit	42.38	42.01	41.29
Upper limit	46.22	46.59	47.31

(d)

Following table shows the critical values of t :

		Critical value
df	90%	95%	99%
80	1.664	1.99	2.639

The formula for confidence interval is

Following table shows the requried confidence interval:

	90%	95%	99%
Lower limit	43.1	42.86	42.39
Upper limit	45.5	45.74	46.21