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Student's t-Distribution

Confidence intervals and sample size less than 30

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Types of Distributions

distribution is a description of the possible values of a random variable and the possible occurrences of these values.

A probability distribution is a graph or table that specifies the probablity associated with each possible value that the random variable can assume.

  • Mean Value or Expected Value

    The mean value, or expected value , of a discrete random variable \begin{align*}X\end{align*} is given by the following equation:

    \begin{align*}\mu=E(x)=\sum_{} xp(x)\end{align*}

  • Variance

    The variance of a discrete random variable is given by the following formula:

    \begin{align*}\sigma^2 = \sum_{} (x-\mu)^2 P(x)\end{align*}

  • Standard Deviation

    The square root of the variance, or, in other words, the square root of \begin{align*}\sigma^2\end{align*} , is the standard deviation of a discrete random variable:

    \begin{align*}\sigma=\sqrt{\sigma^2}\end{align*}

A sampling distribution is the probability distribution of a given statistic based on a random sample.

Normal Distribution

A normal distribution is called a bell curve because its shape is comparable to a bell. It has this shape because the majority of the data is concentrated at the middle and slowly decreases symmetrically on either side.  A normal distribution can be described by the mean and standard deviation of the data.

Here is an example of a normal distribution:

License: CC BY-NC 3.0

[Figure1]

The line in the middle represents the mean, and the tick lines along the bottom represent the standard deviation.

Binomial Distribution

Characteristics of a Binomail Probability Distribution

  • The experiment consists of a fixed number of independent trials
  • Each trial results in one of two outcomes: success or failure
  • The probability of success p is the same for each trial
  • The binomial random variable \begin{align*}X\end{align*} is defined as the number of successes in n trials

The binomial distribution is found by calculating the binomial probabilities for k = 0, 1, 2,..., n.  It can be represented in a histogram with k along the x-axis and P(k) along the y-axis.

When n * p > 10 and n(1-p) > 10, the following statements are all true:

  • the normal distribution will provide a good approximation of the binomial distribution
  • Mean for the binomial distribution: μ = np
  • Standard deviation for the binomial distribution: \begin{align*}\sigma=\end{align*} √(np(1-p))

The binompdf function on the TI-83/84 calculator can be used to solve problems involving the probability of a precise number of successes out of a certain number of trials.  The syntax for the binompdf function is binompdf \begin{align*}(n, p, a)\end{align*} , where \begin{align*}n\end{align*} is the number of trials, \begin{align*}p\end{align*} is the probability of success for any particular trial, and \begin{align*}a\end{align*} is the number of successes.

The binomcdf function on the TI-83/84 calculator can be used to solve problems involving the probability of less than or equal to a number of successes out of a certain number of trials.  The syntax for the binomcdf function is binomcdf \begin{align*}(n, p, a)\end{align*}.

Multinomial Distribution

The multinomial distribution gives the probability of any particular combination of numbers of successes for the various categories.

This probability is given by:

\begin{align*}P & = \frac{n!}{n_1!n_2!n_3!\ldots n_k!} \times \left (p_1{^{n_1}} \times p_2{^{n_2}} \times p_3{^{n_3}} \ldots p_k{^{n_k}} \right )\end{align*}

where:

\begin{align*}n\end{align*} is the number of trials.

\begin{align*}p\end{align*} is the probability for each possible outcome.

\begin{align*}k\end{align*} is the number of possible outcomes.

Geometric Distribution

Characteristics of a Geometric Probability Distribution

  • The experiment consists of a sequence of independent trials
  • Each trial results in one of two outcomes: success or failure
  • The probability of success p is the same for each trial
  • The geometric random variable \begin{align*}X\end{align*} is defined as the number of trials until the first success is observed

The geometric distribution is found by calculating the geometric probabilities for k = 0, 1, 2, ..., ∞ .  It can be represented in a histogram with k along the x-axis and P(k) along the y-axis.  Note: As k increases, P(k) approaches 0, so as this happens you can stop your calculations of P(k) for the purpose of creating the distribution.

Mean for the binomial distribution: μ = 1/p

Standard deviation for the binomial distribution: \begin{align*}\sigma=\end{align*} √((1-p)/p2)

Poisson Distribution

In a binomial distribution, if the number of trials, \begin{align*}n\end{align*} , gets larger and larger as the probability of success, \begin{align*}p\end{align*} , gets smaller and smaller, we obtain a Poisson distribution.

Characteristics of a Poisson distribution:

  • The experiment consists of counting the number of events that will occur during a specific interval of time or in a specific distance, area, or volume.
  • The probability that an event occurs in a given time, distance, area, or volume is the same.
  • Each event is independent of all other events.

Poisson Random Variable:

The probability distributionmean, and variance of a Poisson random variable are given as follows:

\begin{align*}p(x) &= \frac{\lambda^x e^{-\lambda}}{x!} \quad x=0, 1, 2, 3, \ldots\\ \mu &= \lambda\\ \sigma^2 &= \lambda\end{align*}

where:

\begin{align*}\lambda=\end{align*} the mean number of events in the time, distance, volume or area

\begin{align*}e=\end{align*} the base of the natural logarithm

Student's T Distribution

When you use \begin{align*}s\end{align*} to estimate \begin{align*}\sigma\end{align*} , you must use \begin{align*}t\end{align*} instead of \begin{align*}z\end{align*} to complete the significance test for a mean.

In calculating the \begin{align*}t-\end{align*} test statistic, we use the formula:

\begin{align*}t=\frac{\bar{x}-\mu_0}{\frac{s}{\sqrt{n}}}\end{align*}

where:

\begin{align*}t\end{align*} is the test statistic and has \begin{align*}n-1\end{align*} degrees of freedom.

\begin{align*}\bar{x}\end{align*} is the sample mean

\begin{align*}\mu_0\end{align*} is the population mean under the null hypothesis.

\begin{align*}s\end{align*} is the sample standard deviation

\begin{align*}n\end{align*} is the sample size

\begin{align*}\frac{s}{\sqrt{n}}\end{align*} is the estimated standard error

df = n - 1

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