When is unknown and the sample is of size n 30, there are two methods for comput

Question

When is unknown and the sample is of size n 30, there are two methods for computing confidence intervals for . 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 , 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 . 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 = 45.0 and sample standard deviation s = 5.6. (a) Compute 90%, 95%, and 99% confidence intervals for 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 using Method 2 with the standard normal distribution. Use s as an estimate for . 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 shorter. 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 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 using Method 2 with the standard normal distribution. Use s as an estimate for . 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? No. 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. 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.

Explanation / Answer

For N = 41

Observe that the intervals based on the Normal assumption, has Lesser (Smaller ) Confidence region.

For N=81

Observe that the different methods have similar confidence interval,

So for large samples it does not matter which method you use

90 95 99 Lower Upper Lower Upper Lower Upper T- Distribution 43.53072 46.46928 43.23336 46.76664 42.63865 47.36135 Normal 43.5657 46.4343 43.28584 46.71416 42.75235 47.24765

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