A T distribution is known as a \"family of distributions\" rather than a single

Question

A T distribution is known as a "family of distributions" rather than a single distribution like the normal distribution. What does this phrase mean and what practice implications does it have for evaluating a computed T value for statistical significance A T distribution is known as a "family of distributions" rather than a single distribution like the normal distribution. What does this phrase mean and what practice implications does it have for evaluating a computed T value for statistical significance

Explanation / Answer

A T distribution is known as a family of distributions as the distribution is dependent on the degrees of freedom. (degrees of freedom = sample size -1 ). Hence the same t value will be closer or farther( in terms of standard deviations) from the mean depending on the degrees of freedom.

Hence, evaluating the statistical significance of a computed T value will depend on the degrees of freedom. This is because the significant t value from the T distribution table is dependent on the degrees of freedom.

For eg:

Let us take two samples from the same population, with sample sizes 20 and 15.

At 95% confidence interval for two tailed test, the significance level t statistic for sample size 20 is ( degrees of freedom 19) = -2.093 and 2.093

Whereas the significance level t statistic for sample size 15 is = -2.145 and 2.145.

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