When is unknown and the sample is of size n 30, there are two methods for comput
ID: 3063641 • Letter: W
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 = 31, with sample mean x = 45.7 and sample standard deviation s = 5.5.
(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.
(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.
(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.
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.
(d) Now consider a sample size of 71. 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.
(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.
(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 shorter.
Yes. 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.
(g) With increased sample size, do the two methods give respective confidence intervals that are more similar?
Explanation / Answer
a)
b)
c)
Yes. The respective intervals based on the t distribution are longer.
d)
e)
f)
Yes. The respective intervals based on the t distribution are longer.
g)
Yes
CI for 90% 95% 99% n 31 31 31 mean 45.7 45.7 45.7 t-value of 90% CI 1.6973 2.0423 2.7500 std. dev. 5.5 5.5 5.5 SE = std.dev./sqrt(n) 0.98783 0.98783 0.98783 ME = t*SE 1.67660 2.01742 2.71653 Lower Limit = Mean - ME 44.02340 43.68258 42.98347 Upper Limit = Mean + ME 47.37660 47.71742 48.41653 90% CI (44.0234 , 47.3766 ) (43.6826 , 47.7174 ) (42.9835 , 48.4165 )Related Questions
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