Unit 5: Inferential Statistics

Hypothesis Testing, t-Test & Significance Levels

Population	The aggregate of all observations of interest (The “Whole”).
Sample	A subset of the population used to infer conclusions about the population.
Parameter	A statistical measure of the Population (e.g., Population Mean $$\mu$$, Population S.D. $$\sigma$$).
Statistic	A statistical measure of the Sample (e.g., Sample Mean $$\bar{x}$$, Sample S.D. $$s$$).

Null Hypothesis ($$H_0$$)	The hypothesis of “no difference” or status quo. Assumed true until proven otherwise.
Alternate Hypothesis ($$H_1$$or$$H_a$$)	The claim we want to test/prove. (e.g., $$\mu \neq \mu_0$$or$$\mu > \mu_0$$).
Significance Level ($$\alpha$$)	The probability of rejecting $$H_0$$ when it is actually true (Type I Error). Common values: 5% ($$0.05$$) or 1% ($$0.01$$).
Degrees of Freedom ($$df$$)	The number of independent values that can vary.

Used when population S.D. ($$\sigma$$) is unknown and sample size is small ($$n < 30$$).

Type of Test	Formula
One Sample t-test (Testing Sample Mean vs Population Mean)	$$t = \frac{\bar{x} – \mu}{s / \sqrt{n}}$$ df: $$n – 1$$ Decision: If $$\|t_{calc}\| > t_{table}$$, Reject $$H_0$$
Two Independent Samples t-test (Comparing means of two groups)	$$t = \frac{\bar{x}_1 – \bar{x}_2}{S_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$$ Pooled Variance ($$S_p$$): $$S_p = \sqrt{\frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1 + n_2 – 2}}$$ df: $$n_1 + n_2 – 2$$

Type of Test

Formula

One Sample t-test
(Testing Sample Mean vs Population Mean)

$$t = \frac{\bar{x} – \mu}{s / \sqrt{n}}$$

df: $$n – 1$$
Decision: If $$|t_{calc}| > t_{table}$$, Reject $$H_0$$

Two Independent Samples t-test
(Comparing means of two groups)

$$t = \frac{\bar{x}_1 – \bar{x}_2}{S_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$$

Pooled Variance ($$S_p$$):
$$S_p = \sqrt{\frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1 + n_2 – 2}}$$

df: $$n_1 + n_2 – 2$$