### Probability

#### Probability Basics

#### Probability Problems

#### Poker Probability

- Probability in stud poker
- Probability of straight
- Probability of flush
- Cards of equal rank
- Probability of no pair

#### Random Variables

#### Discrete Distributions

#### Continuous Distributions

### Probability: Table of Contents

#### Probability Basics

#### Probability Problems

#### Poker Probability

- Probability in stud poker
- Probability of straight
- Probability of flush
- Cards of equal rank
- Probability of no pair

#### Random Variables

#### Discrete Distributions

#### Continuous Distributions

# Student's t Distribution

The **t distribution** (aka, **Studentâ€™s t-distribution**)
is a probability distribution that is used to estimate population
parameters when the sample size is small and/or when the population
variance is unknown.

## Why Use the t Distribution?

According to the central limit theorem, the sampling distribution of a statistic (like a sample mean) will follow a normal distribution, as long as the sample size is sufficiently large. Therefore, when we know the standard deviation of the population, we can compute a z-score, and use the normal distribution to evaluate probabilities with the sample mean.

But sample sizes are sometimes small, and often we do not know the
standard deviation of the population.
When either of these problems occur, statisticians rely on the
distribution of the
**t statistic** (also known as the
**t score**), whose values are given by:

t = [ x - μ ] / [ s / sqrt( n ) ]

where x is the sample mean, μ
is the population mean, s is the standard deviation of the sample, and n is the
sample size. The distribution of the *t* statistic is called the
**t distribution** or the
**Student t distribution**.

The t distribution allows us to conduct statistical analyses on certain data sets that are not appropriate for analysis, using the normal distribution.

## Degrees of Freedom

There are actually many different t distributions. The particular form
of the t distribution is determined by its
**degrees of freedom**. The degrees of freedom refers
to the number of independent observations in a set of data.

When estimating a mean score or a proportion from a single sample,
the number of independent observations is equal to the sample
size minus one. Hence, the distribution of the *t* statistic from
samples of size 8 would be described by a t distribution having
8 - 1 or 7 degrees of freedom. Similarly, a t distribution having
15 degrees of freedom would be used with a sample of size 16.

For other applications, the degrees of freedom may be calculated differently. We will describe those computations as they come up.

## Properties of the t Distribution

The t distribution has the following properties:

- The mean of the distribution is equal to 0 .
- The variance
is equal to
*v*/ (*v*- 2 ), where*v*is the degrees of freedom (see last section) and*v*__>__2. - The variance is always greater than 1, although it is close to 1 when there are many degrees of freedom. With infinite degrees of freedom, the t distribution is the same as the standard normal distribution.

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## When to Use the t Distribution

The t distribution can be used with any statistic having a bell-shaped distribution (i.e., approximately normal). The sampling distribution of a statistic should be bell-shaped if any of the following conditions apply.

- The population distribution is normal.
- The population distribution is symmetric, unimodal, without outliers, and the sample size is at least 30.
- The population distribution is moderately skewed, unimodal, without outliers, and the sample size is at least 40.
- The sample size is greater than 40, without outliers.

The t distribution should *not* be used with small samples from
populations that are not approximately normal.

## Probability and the Student t Distribution

When a sample of size *n* is drawn from a population having a
normal (or nearly normal) distribution, the sample mean can be
transformed into a t statistic, using the equation presented at the
beginning of this lesson. We repeat that equation below:

t = [ x - μ ] / [ s / sqrt( n ) ]

where x is the sample mean, μ is the population mean, s is the standard deviation of the sample, n is the sample size, and degrees of freedom are equal to n - 1.

The t statistic produced by this transformation can be associated with
a unique
cumulative probability.
This cumulative probability represents the likelihood of finding
a sample mean less than or equal to x,
given a random sample of size *n*.

The easiest way to find the probability associated with a particular t statistic is to use the T Distribution Calculator, a free tool provided by Stat Trek.

## T Distribution Calculator

The T Distribution Calculator solves common statistics problems, based on the t distribution. The calculator computes cumulative probabilities, based on simple inputs. Clear instructions guide you to an accurate solution, quickly and easily. If anything is unclear, frequently-asked questions and sample problems provide straightforward explanations. The calculator is free. It can found in the Stat Trek main menu under the Stat Tools tab. Or you can tap the button below.

T Distribution Calculator## Notation and t Statistics

Statisticians use t_{α} to
represent the t statistic that has a
cumulative probability
of (1 - α).
For example, suppose we were interested in the t statistic having
a cumulative probability
of 0.95. In this example, α would be equal to (1 - 0.95)
or 0.05. We would refer to the t statistic as t_{0.05}

Of course, the value of t_{0.05} depends on the number
of degrees of freedom. For example,
with 2 degrees of freedom, t_{0.05} is equal to 2.92;
but with 20 degrees of freedom, t_{0.05} is equal
to 1.725.

**Note:** Because the t distribution is symmetric about a mean of zero,
the following is true.

*t*_{α} = -*t*_{1 - alpha}
And
*t*_{1 - alpha} = -*t*_{α}

Thus, if t_{0.05} = 2.92, then t_{0.95} = -2.92.

## Test Your Understanding

**Problem 1**

Acme Corporation manufactures light bulbs. The CEO claims that an average Acme 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?

**Note:** There are two ways to solve this problem, using the T Distribution
Calculator. Both approaches are presented below. Solution A is the traditional
approach. It requires you to compute the t statistic, based on data presented in
the problem description. Then, you use the T Distribution Calculator to find
the probability. Solution B is easier. You simply enter the problem data into
the T Distribution Calculator. The calculator computes a t statistic "behind the
scenes", and displays the probability. Both approaches come up with exactly the
same answer.

**Solution A**

The first thing we need to do is compute the t statistic, based on the following equation:

t = [ x - μ ]
/ [ s / sqrt( n ) ]

t = ( 290 - 300 ) / [ 50 / sqrt( 15) ]

t = -10 / 12.909945 = - 0.7745966

where x is the sample mean, μ is the population mean, s is the standard deviation of the sample, and n is the sample size.

Now, we are ready to use the T Distribution Calculator. Since we know the t statistic, we select "T score" from the Random Variable dropdown box. Then, we enter the following data:

- The degrees of freedom are equal to 15 - 1 = 14.
- The t statistic is equal to - 0.7745966.

The calculator displays 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.

**Solution B:**

This time, we will work directly with the raw data from the problem. We will not compute the t statistic; the T Distribution Calculator will do that work for us. Since we will work with the raw data, we select "Sample mean" from the Random Variable dropdown box. Then, we enter the following data:

- The degrees of freedom are equal to 15 - 1 = 14.
- Assuming the CEO's claim is true, the population mean equals 300.
- The sample mean equals 290.
- The standard deviation of the sample is 50.

The calculator displays the cumulative probability: 0.226. Hence, there is a 22.6% chance that the average sampled light bulb will burn out within 290 days.

**Problem 2**

Suppose scores on an IQ test are normally distributed, with a population mean of 100. Suppose 20 people are randomly selected and tested. The standard deviation in the sample group is 15. What is the probability that the average test score in the sample group will be at most 110?

**Solution:**

To solve this problem, we will work directly with the raw data from the problem. We will not compute the t statistic; the T Distribution Calculator will do that work for us. Since we will work with the raw data, we select "Sample mean" from the Random Variable dropdown box. Then, we enter the following data:

- The degrees of freedom are equal to 20 - 1 = 19.
- The population mean equals 100.
- The sample mean equals 110.
- The standard deviation of the sample is 15.

We enter these values into the T Distribution Calculator. The calculator displays the cumulative probability: 0.996. Hence, there is a 99.6% chance that the sample average will be no greater than 110.

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# Teach Yourself to Shoot

- A scientific approach to marksmanship training.
- For hand-gunners who want to be marksmen.