When ? is unknown and the sample is of size n ? 30, there are two methods for co
ID: 3066765 • 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 = 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
(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
Explanation / Answer
a)
b)
d)
e)
CI for 90% 95% 99% n 41 41 41 mean 45 45 45 t-value of 90% CI 1.6839 2.0211 2.7045 std. dev. 5.6 5.6 5.6 SE = std.dev./sqrt(n) 0.87457 0.87457 0.87457 ME = t*SE 1.47265 1.76758 2.36525 Lower Limit = Mean - ME 43.52735 43.23242 42.63475 Upper Limit = Mean + ME 46.47265 46.76758 47.36525 CI (43.5273 , 46.4727 ) (43.2324 , 46.7676 ) (42.6348 , 47.3652 )Related Questions
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