8.3

Student t Distribution

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Normal distributions are used for populations with known standard deviations. However, for most real-world data, the population standard deviation is unknown.

For such populations, the Student t distribution can estimate the population mean using a sample statistic. In this case, the sample mean is the best point estimate for the population mean.

This distribution can be used for simple random samples from a normally distributed population or when a sample size is greater than 30.

For a normally distributed population, the Student t distribution can be given as shown for all values of size n.

Since the estimated sample size is small, the confidence intervals in this distribution are wider, with larger critical values than the normal distribution. The margin of error can be evaluated using the given formula, which helps compute the confidence interval limits.

The Student t distribution shows sample variability. Although symmetrical, it has a wider distribution and represents greater variability than the normal distribution. It always has a standard deviation greater than 1.

However, as the sample size increases, the Student t distribution comes closer to a normal distribution.

Overview

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The population standard deviation is rarely known in many day-to-day examples of statistics. When the sample sizes are large, it is easy to estimate the population standard deviation using a confidence interval, which provides results close enough to the original value. However, statisticians ran into problems when the sample size was small. A small sample size caused inaccuracies in the confidence interval.

The Student t distribution was developed by William S. Goset (1876–1937) of the Guinness brewery in Dublin, Ireland, to estimate the population standard deviation when the sample sizes were small. The name for this distribution comes from the pen name "Student" used by Gosset.

The Student t distribution is used whenever s is used to estimate σ. If a simple random sample of size n is drawn from an approximately normally distributed with mean μ and unknown population standard deviation σ and t scores are calculated, the t scores follow the Student t distribution with n – 1 degrees of freedom. The t score is interpreted similarly to the z score. It measures how far a value is from its mean μ. For each sample size n, there is a different Student t distribution.

The t score or statistic is given as follows:

t-test formula: t = (x̄−μ)/(s/√n); statistical hypothesis testing; equation diagram.

Properties of the Student t distribution:

The graph for the Student t distribution is similar to the standard normal curve.
The mean for the Student t distribution is zero, and the distribution is symmetric about zero.
The Student t distribution has more probability in its tails than the standard normal distribution because the spread of the t-distribution is greater than that of the standard normal. So the Student t distribution curve is thicker in the tails and shorter in the center than the graph of the standard normal distribution.
The exact shape of the Student t distribution depends on the degrees of freedom. As the degrees of freedom increase, the graph of the Student t distribution becomes more like the graph of the standard normal distribution.
The underlying population of individual observations is assumed to be normally distributed with an unknown population mean μ and unknown population standard deviation σ.

This text is adapted from Section 8.2, A Single Population Mean using the Student's t distribution, Introductory Statistics, Openstax,

Transcript

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Student t Distribution

Population Standard Deviation

Sample Size Small

Degrees of Freedom

T Score Formula

Confidence Interval Estimation

Normal Distribution Assumption

Symmetric Distribution Zero

Tails Probability Greater

Statistical Hypothesis Testing