Inference for Means¶

t-Distribution¶

What Is t-Distribution¶

A continuous probability distribution similar to the normal distribution

• The tails are thicker $\implies$ more chances of getting extreme values

• Degrees of freedom, dof
  • $\uparrow$ dof $\implies$ peak sharper and tails thinner $\implies$ closer to normal distribution
• $\mu = 0$
• $\sigma > 1$, closer to 1 as dof increases

When Is t-Distribution Used¶

• $\sigma$ is unknown and population is approximately normally distributed
• $n < 30$
• t-distribution can be used to
  1. Perform hypothesis tests for $\mu$
  2. Form confidence intervals for $\mu$

Hypothesis Tests for Population Means¶

One-Sample t-test for Mean¶

Test whether the population mean of a normally distributed population has changed

$\sigma$ is unknown

Conditions for One-Sample t-test¶

• If the population is very skewed, a t-test can only be done when $n \geq 30$

Calculate t-value¶

• $t = \dfrac{\overline{x} - \mu}{standard\ error}$
• $standard\ error = \dfrac{s}{\sqrt{n}}$

Calculate dof¶

• $dof = n-1$, if there are multiple $n$, choose the smallest one.

For Differences in Population¶

• $standard\ error = \sqrt{\dfrac{s_A^2}{n_A} + \dfrac{s_B^2}{n_B}}$

t-scores VS z-scores¶

graph LR;

H(Start);
I{Normally distributed?};
H --> I;
I -->|Yes| G;
I -->|No| F;
G{Population variance known?};
G -->|Yes| B(z-score);
G -->|No| C{n < 30?};
C -->|Yes| D(t-score);
C -->|No| B;

F{n ≥ 30?} -->|"Yes (CLT)"| B;
F -->|No| J(Non-parametric tests);

Paired t-test¶

Test whether or not the population means of two pieces of data that are linked are equal by examining the differences between paired data

• The data for a two-sample t-test is from two independent populations

• The data for a paired t-test is linked and come from one population

• Use $d$ for the difference of two measures. For instance, $\mu_d$

Calculate t-value¶

$t = \dfrac{\overline{x_d} - \mu_d}{\frac{s_d}{\sqrt{n}}}$