Pindling Confidence Interval  by Example Series

Given the need to find the confidence interval about a statistical parameter from a sample of size n; the following examples illustrate
the correct methodology to determine the confidence interval:
 

Graphical View of Confidence Intervals

Formula for Confidence Intervals: For mean: ,  For mean ,  For proportion     :, , where

Example 1. Find the 95% confidence interval for the mean of the boiling point of water from a sample of size 32 with mean of 212.5
and standard deviation of 1.2.

Given: The level of confidence is 95% or Probability of 0.95, then  and .

The mean,  and the standard deviation, s or , n=32 (large sample size so use standard normal distribution for estimates).

(from Standard Normal Table, taking the absolute or positive value of z obtained from the table with corresponding
probability of 0.025).
 

Confidence Interval is :,

Solution: The population mean is between: 
 

Lower Limit	Upper Limit
:	:
212.08	212.92

Given: The level of confidence is 98% or Probability of 0.98, then  and .

The mean,  and the standard deviation, s = 0.4 , n =18 (small sample size so use student t distribution for estimates).

(from Student t distribution Table, taking value of t = 0.01 with a d.f. = 20 -1 = 19, t_0.01 = 2.57).
 

Confidence Interval is :,

Solution: The population mean is between: 
 

Lower Limit	Upper Limit
:	:
98.56	99.04

Given: The level of confidence is 90% or Probability of 0.90, then  and .

The mean, , n - x = 100-52>5 (large sample size so use standard normal distribution for estimates).
So q = 1- p = 0.48.

(from Standard Normal Table, taking the absolute or positive value of z obtained from the table with
corresponding probability of 0.05).
 

Confidence Interval is :, , where

Solution: The population mean is between: 
 

Lower Limit	Upper Limit
:	::
0.44	0.60