Question

When σ is unknown and the sample is of size n ≥ 30, there are two methods for computing confidence intervals for μ. (Notice that, When σ is unknown and the sample is of size n < 30, there is only one method for constructing a confidence interval for the mean by using the Student's t distribution with d.f. = n - 1.)

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 = 30, with sample mean x = 45.2 and sample standard deviation s = 5.3.

(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 50. 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.No. 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.

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)

Degree of freedom: df=n-1=30- 1 = 29

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%
29	1.699	2.045	2.756

The formula for confidence interval is

Following table shows the required confidence interval:

	90%	95%	99%
Lower limit	43.56	43.22	42.53
Upper limit	46.84	47.18	47.87

(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 required confidence interval:

	90%	95%	99%
Lower limit	43.61	43.3	42.71
Upper limit	46.79	47.1	47.69

(c)

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

(d)

Following table shows the critical values of t :

		Critical value
df	90%	95%	99%
49	1.677	2.01	2.68

The formula for confidence interval is

Following table shows the required confidence interval:

	90%	95%	99%
Lower limit	43.94	43.69	43.19
Upper limit	46.46	46.71	47.21

(e)

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 required confidence interval:

	90%	95%	99%
Lower limit	43.97	43.73	43.27
Upper limit	46.43	46.67	47.13

(f)

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

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