UNIT 5 • TESTING OF HYPOTHESIS

Student's t-Test
Small Sample Significance

When sample size is small ( $n < 30$ ) and true population variance $\sigma^2$ is unknown, the Student's t-Test evaluates whether an observed mean deviation is statistically significant or merely sampling noise!

STUDENT'S T-STATISTIC FORMULAHEAVY-TAILED T-DISTRIBUTION

t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} \quad \text{with } df = n - 1

1. Sample Standard Dev (s)Unbiased variance estimate:

s = \sqrt{\frac{\sum(x - \bar{x})^2}{n-1}}

2. Degrees of FreedomParameter constraints:

df = n - 1

3. Critical DecisionReject

H_0

|t_{calc}| > t_{\alpha/2, df}

Step-by-Step Worked Example

GUIDED STORY

OUR PROBLEM TO SOLVE

Given the 5 sample observations:

[12, 15, 11, 14, 13]

Test the null hypothesis $H_0: \mu = 10$ at significance level $\alpha = 0.05$ !

STEP 01 • COMPUTE SAMPLE MEAN & STANDARD DEVIATION

Calculate sample mean $\bar{x}$ and sample standard deviation $s$ :

\bar{x} = \frac{12 + 15 + 11 + 14 + 13}{5} = \frac{65}{5} = 13.0

s^2 = \frac{(-1)^2 + 2^2 + (-2)^2 + 1^2 + 0^2}{5 - 1} = \frac{10}{4} = 2.5 \implies s = \sqrt{2.5} \approx 1.5811

SE = \frac{s}{\sqrt{n}} = \frac{1.5811}{\sqrt{5}} \approx 0.7071

STEP 02 • COMPUTE T-STATISTIC & THRESHOLD

Calculate test statistic with $df = 4$ :

t_{\text{calc}} = \frac{13.0 - 10}{0.7071} = 4.2426

|t_{\text{calc}}| = 4.2426 > t_{0.025, 4} = 2.776

FINAL RESULT

Decision: Reject H₀ (Mean is Significantly Greater)

\boxed{\text{Reject } H_0: |t_{\text{calc}}| = 4.2426 > 2.776}

Because calculated $|t_{calc}| = 4.2426$ exceeds the critical value $2.776$ at $df = 4$ , we reject the null hypothesis and conclude the mean is significantly greater than 10!

LABORATORY

Interactive Live Calculator & Visualizer

HYPOTHESIS TESTING LABORATORY

Student's t-Test Engine (One-Sample)

PreviousZ-Test (Significance)

Unit 5•26 of 28

NextChi-Square Test

∑

f(x)∫dxπ

NETZ Engine

Evaluating numerical algorithm

UNIT 5 • TESTING OF HYPOTHESIS

Student's t-Test
Small Sample Significance

STUDENT'S T-STATISTIC FORMULAHEAVY-TAILED T-DISTRIBUTION

t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} \quad \text{with } df = n - 1

1. Sample Standard Dev (s)Unbiased variance estimate:

s = \sqrt{\frac{\sum(x - \bar{x})^2}{n-1}}

2. Degrees of FreedomParameter constraints:

df = n - 1

3. Critical DecisionReject

H_0

|t_{calc}| > t_{\alpha/2, df}

Step-by-Step Worked Example

GUIDED STORY

OUR PROBLEM TO SOLVE

Given the 5 sample observations:

[12, 15, 11, 14, 13]

Test the null hypothesis $H_0: \mu = 10$ at significance level $\alpha = 0.05$ !

STEP 01 • COMPUTE SAMPLE MEAN & STANDARD DEVIATION

Calculate sample mean $\bar{x}$ and sample standard deviation $s$ :

\bar{x} = \frac{12 + 15 + 11 + 14 + 13}{5} = \frac{65}{5} = 13.0

s^2 = \frac{(-1)^2 + 2^2 + (-2)^2 + 1^2 + 0^2}{5 - 1} = \frac{10}{4} = 2.5 \implies s = \sqrt{2.5} \approx 1.5811

SE = \frac{s}{\sqrt{n}} = \frac{1.5811}{\sqrt{5}} \approx 0.7071

STEP 02 • COMPUTE T-STATISTIC & THRESHOLD

Calculate test statistic with $df = 4$ :

t_{\text{calc}} = \frac{13.0 - 10}{0.7071} = 4.2426

|t_{\text{calc}}| = 4.2426 > t_{0.025, 4} = 2.776

FINAL RESULT

Decision: Reject H₀ (Mean is Significantly Greater)

\boxed{\text{Reject } H_0: |t_{\text{calc}}| = 4.2426 > 2.776}

Because calculated $|t_{calc}| = 4.2426$ exceeds the critical value $2.776$ at $df = 4$ , we reject the null hypothesis and conclude the mean is significantly greater than 10!

LABORATORY

Interactive Live Calculator & Visualizer

HYPOTHESIS TESTING LABORATORY

Student's t-Test Engine (One-Sample)

PreviousZ-Test (Significance)

Unit 5•26 of 28

NextChi-Square Test

Student's t-Test Small Sample Significance

Step-by-Step Worked Example

Decision: Reject H₀ (Mean is Significantly Greater)

Interactive Live Calculator & Visualizer

Student's t-Test Engine (One-Sample)

NETZ Engine

Student's t-Test Small Sample Significance

Step-by-Step Worked Example

Decision: Reject H₀ (Mean is Significantly Greater)

Interactive Live Calculator & Visualizer

Student's t-Test Engine (One-Sample)

Student's t-Test
Small Sample Significance

Student's t-Test
Small Sample Significance