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

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 sas an estimate for ?, and then use the standard normal distribution.
This method is based on the fact that for large samples, sis a fairly good approximation for ?. Also, for large n, the critical values for the Student's tdistribution approach those of the standard normal distribution.

(a) Now consider a sample size of 71, with sample mean x= 44.2, and sample standard deviation s = 6.2. Compute 90%, 95%, and 99% confidence intervals for ? using Method 1 with a Student's tdistribution. 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 sas an estimate for ?. Round endpoints to two digits after the decimal.

90% 95% 99% lower limit     upper limit    

Explanation / Answer

a)here for (n-1=70) degree of freedom and 90,95 and 99% confidence inteval ; critical t are 1.667 ; 1.994 and 2.648

as confidence interval =sample mean -/+ t*std deviation/sqrt(n)

hence

b)

for 90,95 and 99% confidence inteval ; critical z are 1.645 ; 1.96 and 2.58

confidence limit =sample mean -/+ z*std deviation/sqrt(n)

90% 95% 99% lower limit 42.97 42.73 42.25 upper limit 45.43 45.67 46.15

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