Confidence interval when the Standard Deviation of population is known

Whenever we are going to predict a population parameter using sample data, it is always affected by some error. And this error is known as “Margin of Error”.

So the Interval Estimation can be expressed as:
 

 

 

Confidence Interval is dependent on:

1. Sample Size: More the sample size, lesser is the margin of error and narrower will be the confidence interval
   
2. Variability of Population: More the Variability in population Data, more will be the standard deviation of population and more will be the Margin of Error
   
3. Confidence Level: Higher the confidence level, wider will be the confidence interval
     

 

  

The 95% confidence level is the middle portion in the above figure and the leftover section on the left and right side is 5% combined which is 2.5% on each side.
 

Hence when we are referring to the z-distribution table, we need to look at the value 95 + 2.5 = 97.5 to get the critical value of 95% confidence level.
 

Look at the row and column value for the cell 0.975.

Row value is 1.9

Column value is 0.06
 

Hence Critical value will be 1.9 + 0.06 = 1.96 to get 95% confidence level

 

 

  

 

  

Problem:

We are given with the Standard Deviation of population weight to be 4 Kgs, and a sample of 100 people is chosen and their average weight is 62 kgs. What is the Standard Error, Margin of Error and Confidence interval for Confidence Level 95%?

Let us write what is given:

  Standard Deviation of population = 4

  Sample Size = 100

  Sample Mean = 62

  Confidence Level = 95

 

Standard Error = 4 / Square_root(100) = 4 / 10 = 0.4
 

For 95% confidence level, critical value is 1.96

Margin of Error = 1.96 x Standard Error

Margin of Error = 0.784
 

Confidence Interval = Sample mean +- Margin of Error
 

Lower bound of Confidence Interval = 62 - 0.784 = 61.216

Upper bound of Confidence Interval = 62 + 0.784 = 62.784
 

Hence the confidence interval will be [61.216, 62.784] with 95% of confidence level.
   

This means that there is 95% probability that the estimated confidence interval [61.216, 62.784] will contain the true population mean.