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 = 36, with sample mean x = 44.8 and sample standard deviation s = 4.8.

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.

Explanation / Answer

for method 2 we will use z distribution :

std error of mean=std deviation/(n)1/2 =4.8/(36)1/2 =0.8

for 90% CI;

z =1.6449

hence lower limit =sample mean -z*Std error =43.48

upper limit =sample mean -z*Std error =46.12

for 95% CI;

z =1.96

hence lower limit =sample mean -z*Std error =43.23

upper limit =sample mean -z*Std error =46.37

for 99% CI;

z =2.5758

hence lower limit =sample mean -z*Std error =42.74

upper limit =sample mean -z*Std error =46.86

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