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¶

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