Confidence interval when the Standard Deviation of population is unknown

It is the case where we need to come up with the interval estimation while using the details of the sample only. We don’t have any information available with us for the population.

When to use T-distribution table:

  - When the standard deviation of population is not known

  - When the sample size is less than 30

 

Because of the above 2 reasons, instead of referring to the z-distribution table, we refer to the T-distribution table.

 

The variance in a t-distribution is estimated based on the degrees of freedom of the data set (total number of observations minus 1).

 

We refer to T table while using two values as input parameter, Confidence level and degree of freedom.

As in this case, we do not know Population Standard Deviation, so we use Degrees of Freedom as our base for estimation. For the different values of the Degree of Freedom, we will get Different critical values.

 

Degree of Freedom = Sample Size - 1 

                                = 25 - 1 

                                = 24

 

Confidence Level = 90

 

 

For Sample Size = 25, Degree of Freedom will be 24 and to get 90% confidence level, critical value will be 1.711

 

For Sample Size = 25, Degree of Freedom will be 24 and to get 95% confidence level, critical value will be 2.064

 

 

Problem: We have given a sample of 15 people and their average weight is 62 kgs and standard deviation of the sample is 4. What is the Standard Error, Margin of Error and Confidence interval for Confidence Level 95%?

Let us write what is given: 

   Standard Deviation of sample = 4

   Sample Size = 15

   Degree of Freedom = 15 - 1 = 14

   Sample Mean = 62

   Confidence Level = 95

 

   Standard Error = 4 / Square_root(15) = 1.03

 

   For 95% confidence level for degree of freedom 14, the critical value is 2.145

 

   Margin of Error = 2.145 x Standard Error

   Margin of Error = 2.21

 

   Confidence Interval = Sample mean +- Margin of Error

  

   Lower bound of Confidence Interval = 62 - 2.21 = 59.79

   Upper bound of Confidence Interval = 62 + 2.21 = 64.21

 

   Hence the confidence interval will be [59.79, 64.21] with 95% of confidence level.

 

   This means that there is 95% probability that the estimated confidence interval [59.79, 64.21] will contain the true population mean.