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Difference between Z Distribution and T Distribution
The Z distribution is a special case of the normal distribution with a mean of 0 and standard deviation of 1. The t-distribution is similar to the Z-distribution, but is sensitive to sample size and is used for small or moderate samples when the population standard deviation is unknown. At large samples, the z and t samples are very similar.
The t-statistic is used to test hypotheses about an unknown population mean u when the value of "$ \sigma $" is unknown. The formula for the t statistic has the same structure as the z-score formula, except that the t statistic uses the estimated standard error in the denominator. The only difference between the t formula and the z-score formula is that the z-score uses the actual population variance, "$ \sigma^2 $" (or the standard deviation) and the t formula uses the corresponding sample variance (or standard deviation) when the population value is not known. Simply put, the basic difference between these two is that the t statistic uses sample variance (s 2) and the z-score uses the population variance "$ \left( \sigma^2 \right) $". To determine how well a t-statistic approximates a z-score, we must determine how well the sample variance approximates the population variance. Basically, for small samples, the t-statistic is used.
"$ \begin{align} Z = \dfrac{X - \mu}{\sigma} &&& t = \dfrac{\bar x - \mu_0}{\frac{s}{\sqrt{n}}}, \end{align}$"
The mean must be known prior to computing the sample variance. This places a restriction on sample variability such that only n-1 scores in a sample are free to vary. The value n-1 is called the degrees of freedom (df) for the sample variance. Degrees of freedom describe the number of scores in a sample that are free to vary. Because the sample mean places a restriction on the value of one score in the sample, there are n-1 degrees of freedom for the sample.
The t Distribution and Shape of the t Distribution
The t-distribution will approximate a normal distribution in the same way that a t statistic approximates a z-score. How well a t distribution approximates a normal distribution is determined by degrees of freedom (df). The greater the sample size (n) is, the larger the degrees of freedom (n-1) are, and the better the t-distribution approximates the normal distribution. The exact shape of a t distribution changes with df. As df gets very large, the t distribution gets closer in shape to a normal z-score distribution. Distributions of t are bell-shaped and symmetrical and have a mean of zero. The t-distribution tends to be flatter and more spread out, whereas the normal z-distribution has more of a central peak.