Muhammad Imdad Ullah

The post is about the use of t Distribution in Statistics. The t distribution, also known as the Student’s t-distribution, is a probability distribution used to estimate population parameter(s) when the sample size is small or when the population variance is unknown. The t distribution is similar to the normal bell-shaped distribution but has heavier tails. This means that it gives a lower probability to the center and a higher probability to the tails than the standard normal distribution.

The t distribution is particularly useful as it accounts for the extra variability that comes with small sample sizes, making it a more accurate tool for statistical analysis in such cases.

The following are the commonly used situations in which t distribution is used:

Use of t Distribution: Confidence Intervals

The t distribution is widely used in constructing confidence intervals. In most of the cases, The width of the confidence intervals depends on the degrees of freedom (sample size – 1):

1. Confidence Interval for One Sample Mean
   $$\overline{X} \pm t_{\frac{\alpha}{2}} \left(\frac{s}{\sqrt{n}} \right)$$
   where $t_{\frac{\alpha}{2}}$ is the upper $\frac{\alpha}{2}$ point of the t distribution with $v=n-1$ degrees of freedom and $s^2$ is the unbiased estimate of the population variance obtained from the sample, $s^2 = \frac{\Sigma (X_i-\overline{X})^2}{n-1} = \frac{\Sigma X^2 – \frac{(\Sigma X)^2}{n}}{n-1}$
2. Confidence Interval for Difference between Two Independent Samples MeanL
   Let $X_{11}, X_{12}, \cdots, X_{1n_1}$ and $X_{21}, X_{22}, \cdots, X_{2n_2}$ be the random samples of size $n_1$ and $n_2$ from normal population with variances $\sigma_1^2$ and $\sigma_2^2$, respectively. Let $\overline{X}_1$ and $\overline{X}_2$ be the respectively sample means. The confidence interval for the difference between two population mean $\mu_1 – \mu_2$ when the population variances $\sigma_1^2$ and $\sigma_2^2$ are unknown and the sample sizes $n_1$ and $n_2$ are small (less than 30) is
   $$(\overline{X}_1 – \overline{X}_2 \pm t_{\frac{\alpha}{2}}(S_p)\sqrt{\frac{1}{n_1} + \frac{1}{n_2}}$$
   where $S_p = \frac{(n_1 – 1)s_1^2 + (n_2-1)s_2^2}{n_1-n_2-2}$ (Pooled Variance), where $s_1^2$ and $s_2^2$ are the unbiased estimates of population variances $\sigma_1^2$ and $\sigma_2^2$, respectively.
3. Confidence Interval for Paired Observations
   The confidence interval for $\mu_d=\mu_1-\mu_2$ is
   $$\overline{d} \pm t_{\frac{\alpha}{2}} \frac{S_d}{\sqrt{n}}$$
   where $\overline{d}$ and $S_d$ are the mean and standard deviation of the differences of $n$ pairs of measurements and $t_{\frac{\alpha}{2}}$ is the upper $\frac{\alpha}{2}$ point of the distribution with $n-1$ degrees of freedom.

Use of t Distribution: Testing of Hypotheses

The t-tests are used to compare means between two groups or to test if a sample mean is significantly different from a hypothesized population mean.

1. Testing of Hypothesis for One Sample Mean
   It compares the mean of a single sample to a known population mean when the population standard deviation is known,
   $$t=\frac{\overline{X}-\mu}{\frac{s}{\sqrt{n}}}$$
2. Testing of Hypothesis for Difference between Two Population Means
   For two random samples of sizes $n_1$ and $n_2$ drawn from two normal population having equal variances ($\sigma_1^2 = \sigma_2^2 = \sigma^2$), the test statistics is
   $$t=\frac{\overline{X}_1 – \overline{X}_2}{S_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$$
   with $v=n_1+n_2-2$ degrees of freedom.
3. Testing of Hypothesis for Paird/Dependent Observations
   To test the null hypothesis ($\mu_d = \mu_o$) the statistics is
   $$t=\frac{\overline{d} – d_o}{\frac{s_d}{\sqrt{n}}}$$
   with $v=n-1$ degrees of freedom.
4. Testing the Coefficient of Correlation
   For $n$ pairs of observations (X, Y), the sample correlation coefficient, the test of significance (testing of hypothesis) for the correlation coefficient is
   $$t=\frac{r\sqrt{n-2}}{\sqrt{1-r^2}}$$
   with $v=n-2$ degrees of freedom.
5. Testing the Regression Coefficients
   The t distribution is used to test the significance of regression coefficients in linear regression models. It helps determine whether a particular independent variable ($X$) has a significant effect on the dependent variable ($Y$). The regression coefficient can be tested using the statistic
   $$t=\frac{\hat{\beta} – \beta}{\sqrt{SE_{\hat{\beta}}}}$$
   where $SE_{\hat{\beta}} = \frac{S_{Y\cdot X}}{\sqrt{\Sigma (X-\overline{X})^2}}=\frac{\sqrt{\frac{\Sigma Y^2 – \hat{\beta}_o \Sigma X – \hat{\beta}_1 \Sigma XY }{n-2} } }{S_X \sqrt{n-1}}$

The t distribution is a useful statistical tool for data analysis as it allows the user to make inferences/conclusions about population parameters even when there is limited information about the population.

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Frequently Asked Questions about the Use of t Distribution

• What is t distribution?
• Discuss what type of confidence intervals can be constructed by using t distribution.
• Discuss what type of hypothesis testing can be performed by using t distribution.
• How does the t distribution resemble the normal distribution?
• What is meant by small sample size and unknown population variance?