T-Distribution

T-Distribution is a continuous probability distribution. It is used when sample sizes are smaller than the normal distribution, say less than 30. This method identifies the disparity between the sample and population means when the population standard deviation is unknown.

It is used in statistics to determine extreme confidence interval values for a normal distribution with small sample size. Just like normal distributions, the T-Distribution forms a symmetric bell-shaped curve. But T-curves have fatter tails than a normal distribution curve; this depicts extreme confidence interval values.

William Sealy Gosset introduced the T-Distribution in statistics as a probability analysis method in 1908. It is applied to cases with small sample sizes and obscure standard deviation (population). 

The following properties of T-Distribution differentiate it from the other kinds of probability distributions:

• The T-Distribution or student T-Distribution forms a symmetric bell-shaped curve with fatter tails.
• Its mean comes out to be zero.
• The value of the distribution ranges between -∞ and ∞.
• Its variance is computed as v/(v-2). Here, v ≥ 2 and ‘v’ denotes the degree of freedom: ‘Var (t) = v/(v -2)’.
• The variance values are higher than 1 when the degree of freedom is infinite; it provides a value close to the standard normal distribution.
• Compared to standard normal distributions, students' T-Distribution is highly dispersed. However, in the case of larger sample sizes, i.e., n ≥ 30, it resembles a normal distribution.

Let us have a look at the T-Distribution graph:

Although it is a widely used method, it is criticized for its inaccuracy (in certain cases). Also, when larger sample sizes, normal distribution tends to be a better option.

This method involves the computation of two values:

#1 - T-Score

The formula used to determine the value of T-Distribution is as follows:

Here,

• x̄ is the sample mean;
• μ is the population mean;
• s is the standard deviation;
• n is the sample size.

#2 - Degree of Freedom

Variance is derived using the degree of freedom for the given data series. It is computed as sample size minus 1:

df = n - 1

Here,

• ‘df’ is the degree of freedom;
• ‘n’ is the sample size.

The value derived from the above formula is the t-score. Then, the value of the t-score and the degree of freedom are used to determine the p-value or probability using the T-Distribution table. This way, the chance of getting the desired outcome is determined. 

Alternatively, T-distribution calculators from the internet can be used to derive the results.

ABC Poultry Farms supplies eggs. The company claims its eggs remain fresh for five days if refrigerated. An analyst samples 25 eggs to test this claim. The average freshness of eggs was 4.5 days, with a standard deviation of a day. If the company's claim is true, find the probability of all selected eggs lasting about 4.5 days.

Solution: 
Given:

• x̄ = 4.5 days
• μ = 5 days
• s = 1 day
• n = 25

Therefore,
t = (x̄-µ)/(s/√n)

t = (4.5 – 5)/(1/√25)

t = -0.5/0.2 = -2.5

Since the minus sign is irrelevant here, we get t = 2.5.

Degree of Freedom (df) = n – 1

df = 25 – 1 = 24

Thus, according to the t-test, the probability (p-value) of eggs not lasting for more than 4.5 days is 0.01965418.

Note: To find the p-value, we have substituted the values of t-score and degree of freedom into an online calculator to get the result: 0.01965418.