What Is Student’s T – Distribution?

(1) What Is Student’s T – Distribution?

Student’s t-distribution, also known as the t-distribution, is a continuous probability distribution that is used in statistics for making inferences about the population mean when the sample size is small or when the population standard deviation is unknown.
It is similar to the standard normal distribution (Z-distribution), but it has heavier tails.
The t-distribution is used instead of the normal distribution when you have small samples.
The larger the sample size, the more the t distribution looks like the normal distribution.\
In fact, for sample sizes larger than 20 (e.g. more degrees of freedom), the distribution is almost exactly like the normal distribution.

where,
t = The t-score,
x̄ = sample mean,
μ = population mean,
s = standard deviation of the sample,
N = sample size

Student’s t Distribution is used when
- The sample size is 30 or less than 30.
- The population standard deviation(σ) is unknown.
- The population distribution must be unimodal and skewed.

The t-distribution is a type of normal distribution that is used for smaller sample sizes.
Normally-distributed data form a bell shape when plotted on a graph, with more observations near the mean and fewer observations in the tails.
The t-distribution is used when data are approximately normally distributed, which means the data follow a bell shape but the population variance is unknown.
The variance in a t-distribution is estimated based on the degrees of freedom of the data set (total number of observations minus 1).
It is a more conservative form of the standard normal distribution, also known as the z-distribution.
This means that it gives a lower probability to the center and a higher probability to the tails than the standard normal distribution.

The CEO of light bulbs manufacturing company claims that an average light bulb lasts 300 days.
A researcher randomly selects 15 bulbs for testing. The sampled bulbs last an average of 290 days, with a standard deviation of 50 days.
If the CEO’s claim were true, what is the probability that 15 randomly selected bulbs would have an average life of no more than 290 days?

x̄ = is the sample mean,
μ = is the population mean, s is the standard deviation of the sample, and n is the sample size.
Using the formula:

Since we will work with the raw data, we select “Sample mean” from the Random Variable dropdown box.

The cumulative probability: 0.226.
Hence, if the true bulb life were 300 days, there is a 22.6% chance that the average bulb life for 15 randomly selected bulbs would be less than or equal to 290 days

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.

Given:

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.

Mahindra Company claims that the ‘THAR’ car gives average milage of 20km/l.
I can test this claim by examining 10 ‘THAR’ cars and can find it’s milage with some standard deviation.
I can also find out the probability of happening that by using T-Distribution.