Hypothesis testing is a statistical method used to determine whether there is enough evidence from sample data to support a claim about a population.

Null & Alternative Hypotheses
Hypothesis Testing Workflow
One-sample t-test
Two-sample t-test
p-value
Significance Level (α)
Type I & Type II Errors
Confidence Intervals
Choosing the Correct Test
Complete Python Examples
Interview Questions
Summary

1. Null & Alternative Hypotheses

A hypothesis is an assumption about a population parameter. There are always two hypotheses.

Null Hypothesis (H₀)

The null hypothesis assumes no effect, no difference, or no relationship. Examples:

Average salary = ₹50,000
New medicine is not better than old medicine.
Two algorithms have equal accuracy.

Example:

H₀: μ = 50

where

μ = population mean

Alternative Hypothesis (H₁ or Hₐ)

The alternative hypothesis represents what we want to prove. Examples

H₁: μ ≠ 50

H₁: μ > 50

H₁: μ < 50

Types of Alternative Hypotheses

Two-tailed

Testing for any difference.

H₀: μ = 50
H₁: μ ≠ 50

Right-tailed

Testing if value is greater.

H₀: μ ≤ 50
H₁: μ > 50

Left-tailed

Testing if value is smaller.

H₀: μ ≥ 50
H₁: μ < 50

Hypothesis Testing Workflow

State hypotheses
        ↓
Collect sample
        ↓
Choose statistical test
        ↓
Calculate test statistic
        ↓
Compute p-value
        ↓
Compare with α
        ↓
Reject or Fail to Reject H₀

2. One-Sample t-test

Used when comparing one sample mean against a known population value.

Example

A company claims average battery life is 10 hours. Sample:

[10.5, 9.8, 10.2, 10.4, 9.9]

Question: Is the actual average different from 10?

Formula

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

where

x̄ = sample mean
μ = population mean
s = sample standard deviation
n = sample size

Python Example

from scipy.stats import ttest_1samp

battery = [10.5, 9.8, 10.2, 10.4, 9.9]

t_stat, p_value = ttest_1samp(battery, popmean=10)

print("t statistic:", t_stat)
print("p-value:", p_value)

Interpretation

p < 0.05

Reject H₀ Otherwise Fail to reject H₀.

3. Two-Sample t-test

Used to compare means of two independent groups. Example Do students using Method A score differently than students using Method B?

Sample Method A

[78, 82, 85, 88, 90]

Method B

[72, 75, 79, 80, 81]

Python

from scipy.stats import ttest_ind

group1 = [78,82,85,88,90]
group2 = [72,75,79,80,81]

t_stat, p = ttest_ind(group1, group2)

print(t_stat)
print(p)

Equal Variance Assumption

If variances are unequal

ttest_ind(group1, group2, equal_var=False)

This performs Welch’s t-test.

Difference Between One-Sample and Two-Sample t-test

Feature	One Sample	Two Sample
Number of groups	1	2
Compare against	Known value	Another group
Example	Mean = 50?	Group A vs Group B
Function	`ttest_1samp()`	`ttest_ind()`

4. p-value

The p-value measures how likely the observed data would occur if the null hypothesis were true. It is not the probability that the null hypothesis is true.

Interpretation

p-value	Decision
< 0.01	Very strong evidence against H₀
< 0.05	Strong evidence against H₀
> 0.05	Not enough evidence to reject H₀

Example

α = 0.05

p = 0.03

Since

0.03 < 0.05

Reject H₀.

Another Example

p = 0.18

Cannot reject H₀.

5. Significance Level (α)

The significance level is the threshold for deciding whether to reject the null hypothesis. Common choices

10
05
01

Most commonly

α = 0.05

Meaning Accept a 5% chance of incorrectly rejecting a true null hypothesis.

Decision Rule

If p ≤ α

Reject H₀

Otherwise

Fail to Reject H₀

6. Type I & Type II Errors

Type I Error (False Positive)

Rejecting a true null hypothesis. Reality:

Medicine doesn't work.

Conclusion:

Medicine works.

Wrong conclusion. Probability

α

Type II Error (False Negative)

Failing to reject a false null hypothesis. Reality

Medicine works.

Conclusion

Medicine doesn't work.

Probability

β

Statistical Power

Power

Power = 1 - β

Higher power means a better chance of detecting a true effect.

Error Table

Reality	Decision	Result
H₀ True	Reject	Type I Error
H₀ True	Fail to Reject	Correct
H₀ False	Reject	Correct
H₀ False	Fail to Reject	Type II Error

7. Confidence Intervals

A confidence interval gives a range where the true population parameter is likely to lie. Example

95% CI

(48.2, 52.6)

Interpretation: Using this method repeatedly, about 95% of the intervals constructed would contain the true population mean.

Formula

Confidence Interval = x̄ ± t* × (s / √n)

Where:

x̄ = Sample mean
t* = Critical t-value (depends on confidence level and degrees of freedom)
s = Sample standard deviation
n = Sample size

Python Example

import numpy as np
from scipy import stats

data = [10,11,9,10,12,11,9]

confidence = 0.95

mean = np.mean(data)
sem = stats.sem(data)

interval = stats.t.interval(
    confidence,
    len(data)-1,
    loc=mean,
    scale=sem
)

print(interval)

Interpretation

If the confidence interval contains the hypothesized value

Example

95% CI

(48, 53)

Hypothesized mean = 50

Since

50 lies inside interval

Fail to reject H₀ at the 5% significance level.

Choosing the Correct Test

Situation	Test
Compare one sample with known mean	One-sample t-test
Compare two independent groups	Two-sample t-test
Compare paired measurements	Paired t-test
Compare more than two groups	ANOVA

Complete Python Example

import numpy as np
from scipy.stats import ttest_1samp, ttest_ind

# ----------------------------
# One Sample t-test
# ----------------------------

sample = [52, 49, 51, 48, 50, 52, 53]

t, p = ttest_1samp(sample, popmean=50)

print("One Sample Test")
print("t =", t)
print("p =", p)

# ----------------------------
# Two Sample t-test
# ----------------------------

group1 = [75,80,82,78,77]
group2 = [70,72,71,68,74]

t, p = ttest_ind(group1, group2)

print("\nTwo Sample Test")
print("t =", t)
print("p =", p)

Interview Questions

What is the null hypothesis?

A statement that assumes there is no difference or no effect.

What does a p-value represent?

The probability of observing results at least as extreme as those obtained if the null hypothesis is true.

Why use a t-test?

To compare means when the population standard deviation is unknown.

Difference between α and p-value?

α is chosen before the experiment (decision threshold).
p-value is computed from the observed sample data.

What is a 95% confidence interval?

An interval produced by a method that, over many repeated samples, would contain the true population parameter about 95% of the time.

Summary

Topic	Key Idea
Null Hypothesis	No effect or difference
Alternative Hypothesis	Effect or difference exists
One-Sample t-test	Compare one sample with known mean
Two-Sample t-test	Compare two independent means
p-value	Evidence against H₀
Significance Level (α)	Decision threshold
Type I Error	False Positive
Type II Error	False Negative
Confidence Interval	Plausible range for the population parameter
Power	Probability of detecting a true effect (`1 - β`)

Key Takeaways

Start by defining H₀ and H₁.
Choose the correct statistical test based on the data.
Compare the p-value with α to make a decision.
Understand the risks of Type I and Type II errors.
Use confidence intervals to estimate plausible values for the population parameter.
Report both the test result and the confidence interval for a more complete statistical interpretation.

​Table of Contents

​1. Null & Alternative Hypotheses

​Null Hypothesis (H₀)

​Alternative Hypothesis (H₁ or Hₐ)

​Types of Alternative Hypotheses

​Two-tailed

​Right-tailed

​Left-tailed

​Hypothesis Testing Workflow

​2. One-Sample t-test

​Example

​Formula

​Python Example

​Interpretation

​3. Two-Sample t-test

​Python

​Equal Variance Assumption

​Difference Between One-Sample and Two-Sample t-test

​4. p-value

​Interpretation

​5. Significance Level (α)

​6. Type I & Type II Errors

​Type I Error (False Positive)

​Type II Error (False Negative)

​Statistical Power

​Error Table

​7. Confidence Intervals

​Formula

​Python Example

​Interpretation

​Choosing the Correct Test

​Complete Python Example

​Interview Questions

​What is the null hypothesis?

​What does a p-value represent?

​Why use a t-test?

​Difference between α and p-value?

​What is a 95% confidence interval?

​Summary

​Key Takeaways

Table of Contents

1. Null & Alternative Hypotheses

Null Hypothesis (H₀)

Alternative Hypothesis (H₁ or Hₐ)

Types of Alternative Hypotheses

Two-tailed

Right-tailed

Left-tailed

Hypothesis Testing Workflow

2. One-Sample t-test

Example

Formula

Python Example

Interpretation

3. Two-Sample t-test

Python

Equal Variance Assumption

Difference Between One-Sample and Two-Sample t-test

4. p-value

Interpretation

5. Significance Level (α)

6. Type I & Type II Errors

Type I Error (False Positive)

Type II Error (False Negative)

Statistical Power

Error Table

7. Confidence Intervals

Formula

Python Example

Interpretation

Choosing the Correct Test

Complete Python Example

Interview Questions

What is the null hypothesis?

What does a p-value represent?

Why use a t-test?

Difference between α and p-value?

What is a 95% confidence interval?

Summary

Key Takeaways