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T Distribution

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Definition:

The t-distribution, also known as Student’s t-distribution, is a probability distribution that describes the sampling distribution of the t-statistic, which is a standardized measure of the difference between a sample mean and the population mean, divided by the sample standard deviation.

Key Properties:

  • Central tendency: The t-distribution has a mean of 0.
  • Symmetry: The t-distribution is symmetrical about 0.
  • Unimodality: The t-distribution is unimodal, meaning it has only one mode at 0.
  • Skewness: The t-distribution has a skewness of 0.
  • Kurtosis: The t-distribution has a kurtosis of 0.
  • Degrees of freedom: The t-distribution is defined by a number of degrees of freedom, which is the number of observations in the sample minus 1.
  • Tail behavior: The t-distribution has heavy tails, meaning that it has a high probability of extreme values.

Uses:

  • T-tests: The t-distribution is used to perform t-tests, which are used to compare the means of two or more groups.
  • Confidence intervals: The t-distribution is used to calculate confidence intervals for the population mean.
  • Hypothesis testing: The t-distribution is used to test hypotheses about the population mean.

Formula:

The t-distribution is given by the following formula:

t = (x - ฮผ) / sqrt(s/n)

where:

  • t is the t-statistic
  • x is the sample mean
  • ฮผ is the population mean
  • s is the sample standard deviation
  • n is the number of observations

Note:

The t-distribution is a special case of the Student’s t-distribution, which is a more general family of probability distributions. The Student’s t-distribution is used when the sample size is large and the population standard deviation is unknown.

FAQs

  1. What is the t-distribution in statistics?

    The t-distribution is a probability distribution used to estimate population parameters when the sample size is small and/or the population standard deviation is unknown.

  2. When do we use the t-distribution?

    The t-distribution is used when sample sizes are small (typically under 30) and the population standard deviation is unknown.

  3. What is the t-distribution at 95%?

    The 95% t-distribution critical value depends on the degrees of freedom; for a large sample, it approaches 1.96, similar to the z-distribution.

  4. Why is the t-distribution better for small samples?

    The t-distribution has heavier tails, which provides a more accurate estimate for smaller samples by accounting for variability more effectively.

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