T-Interval (t statistics)

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

T-Distribution

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:

• Random:

The data needs to come from a random sample or randomized experiment.

• Normal:

The sample size is at least 30.

• Independent:

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.

T-score

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_, μ₀ 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

Example

Solve:

T interval for paired data

Refer to article on Khan academy: Making a t interval for paired data

T-table

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