T interval is good for situations where the sample size is small and population standard deviation is unknown.
When the sample size comes to be very small (n?30), the Z-interval for calculating confidence interval becomes less reliable estimate. And here the T-interval comes into place.
Refer to Khan academy: Small sample size confidence intervals
The full name is Student’s t-distribution, which is a _tweaked version of Normal Distribution_.
Refer to wiki: Student?s t-distribution
When the sample size is small, the Normal distribution will no longer be a good fit for estimating the population.So we introduced the tweaked version of Normal Distribution for a small sample sized sampling data, which we called T-distribution.
T-distribution vs. Normal distribution
They have the same centre: Sample Mean.But the tail of t-distribution is ?fatter? than the Normal distribution.
Conditions for a valid T Interval
The conditions we need for inference on one proportion are:
The data needs to come from a random sample or randomized experiment.
The sample size is at least 30.
The independence condition says that when sampling without replacement, we can still treat each observation in the sample as independent as long as we sample less than 10%, percent of the population.
Refer to article: What is the T Score Formula?
A t score is one form of a Standardized Test Statistic (the other you?ll come across in elementary statistics is the z-score). The t score formula enables you to take an individual score and transform it into a standardized form>one which helps you to compare scores.You?ll want to use the t score formula when you don?t know the population standard deviation and you have a small sample (under 30).
The t score formula is:
(x? is the _Sample Mean_, ?? is mean from _null hypothesis_, sx is the _Sample SD_, n is _Sample size_)
Understanding the formula
The statistic – parameter results the DISTANCE from Sample mean to _Population mean_.The Standard Error represents the DISTANCE from Sample SD to _population SD_.=> Therefore, dividing the Distance of mean by Distance of SD will results in a Normalized Distance for mean.
?? Back to previous note on: Standard Error
Formula of T-interval
The difference with Z-interval?s formula is instead of using Z* value, we’ll be using the T* value,and the calculation of Standard Error is different too.
One-sample T interval
T interval for paired data
Refer to article on Khan academy: Making a t interval for paired data