t-distribution for sample means

• Recall that a population distribution $Y$, after standardisation, is normally distributed, where $Y\sim\mathcal{N}(0,1^2)$
• Recall that $\bar Y\sim\mathcal{N}(\mu,(\frac{\sigma}{\sqrt n})^2)$ for drawing indefinite samples, each with size $n$, from $Y$
• Recall that standard error of sample means $\sigma_{\bar Y }=\frac{\sigma}{\sqrt n}$ theoretically
• However, with real data, Z-standardisation is hardly justified to $\bar Y$: $\sigma_{\bar Y}$ is always unknown, the estimate of the standard error of sample means is ****$SE_{\bar Y}=\frac{s}{\sqrt n}$, where $s$ is an estimate of the standard deviation of sample means
• $SE_{\bar Y}$ is a variable, $s$ varies by chance from sample to sample
• Therefore, standardising sample mean $\bar Y$ will not give a normal distribution, instead a Student’s t-distribution with $n-1$ degrees of freedom:

$$ t=\frac{\bar Y - \mu}{SE_{\bar Y}} $$

Plot of the density function for several members of the Student’s t-family characterised by different degrees of freedom

• *t-*distribution resembles standard normal distribution $Z$ such as symmetric around a mean of 0 and roughly bell-shaped, with an important distinction: fatter in the tails towards $\pm\infin$
  • Confidence intervals and hypothesis testing relies on the tails!

Confidence interval for the mean of a normal population

• Similar to a standard normal distribution, t-distributions (with different degrees of freedom) have its own PDF and CDF

• A confidence interval is the probability that a population parameter will fall between a set of values for a certain proportion of times, which is the reverse of critical value $\alpha$ (i.e. $1-\alpha$).

• A 95% confidence interval for the mean of a normal population $\mu$ can be defined from the t-values which bound 95% of the area under the curve of the *t-*distribution for sample means $\bar Y$

  • Those t-values are also then called the 5% critical t-value, $\alpha=0.05$
  • Notated as $t_{\alpha(2),df}$, the $(2)$ indicates that the 5% area is divided between the two tails of the t-distribution
  • Confidence interval = $\bar Y\pm t_{\alpha,df}SE_{\bar Y}$

$t_{0.05(2),4}=\pm2.78$, as 95% of the area under the curve for this *t-*distribution is bounded by [-2.78, 2.78].

• $\alpha$ is usually 0.05 which defines the 95% confidence interval, but it can be any arbitrary number
• It can never be known that if $\mu$ must fall within those critical t-values
• It only implies that in 95% of the samples from the population, the sample mean will fall within the confidence interval