Why hypothesis testing validates claims in SciPy

Introduction

Hypothesis testing helps us check if a claim about data is likely true or just happened by chance.

Checking if a new medicine works better than the old one.

Deciding if a marketing campaign increased sales.

Verifying if students' test scores improved after extra tutoring.

Determining if a machine produces fewer defects after maintenance.

Syntax

SciPy

from scipy import stats

# Example: stats.ttest_1samp(data, popmean)

# data: list or array of sample values
# popmean: the value to test against

Use stats.ttest_1samp for testing if the sample mean differs from a known value.

The test returns a statistic and a p-value to help decide if the claim holds.

Examples

Test if the average of sample is different from 7.

SciPy

from scipy import stats

sample = [5, 6, 7, 8, 9]
popmean = 7
stat, pvalue = stats.ttest_1samp(sample, popmean)
print(stat, pvalue)

Check if sample mean is different from 10.

SciPy

from scipy import stats

sample = [10, 12, 11, 13, 14]
popmean = 10
stat, pvalue = stats.ttest_1samp(sample, popmean)
print(stat, pvalue)

Sample Program

This code tests if the average test score is really 85 or different. It prints the test result and conclusion.

SciPy

from scipy import stats

# Sample data: test scores after tutoring
scores = [82, 85, 88, 80, 87, 84, 89, 81, 86, 88]

# Claim: average score is 85
claimed_mean = 85

# Perform one-sample t-test
statistic, p_value = stats.ttest_1samp(scores, claimed_mean)

print(f"Test statistic: {statistic:.3f}")
print(f"P-value: {p_value:.3f}")

# Decide if claim is valid at 5% significance
if p_value < 0.05:
    print("Reject the claim: data suggests average is not 85.")
else:
    print("Cannot reject the claim: data supports average is 85.")

Important Notes

A small p-value (usually less than 0.05) means the claim is unlikely true.

Hypothesis testing does not prove a claim, it only shows if data supports or rejects it.

Always check assumptions like data being roughly normal for t-tests.

Summary

Hypothesis testing helps check if data supports a claim or not.

It uses a test statistic and p-value to make decisions.

Small p-value means claim is probably false; large p-value means claim could be true.

Practice

1. What does a small p-value in hypothesis testing usually indicate about the claim being tested?

easy

A. The data is not enough to decide

B. The claim is definitely true

C. The claim is probably false and should be rejected

D. The claim is definitely false

Solution

1. Step 1: Understand p-value meaning

   A p-value measures how likely the observed data would occur if the claim (null hypothesis) is true.

2. Step 2: Interpret small p-value

   A small p-value means the observed data is unlikely under the claim, so we reject the claim.

3. Final Answer:

   The claim is probably false and should be rejected -> Option C

4. Quick Check:

   Small p-value = reject claim [OK]

Hint: Small p-value means reject claim, large means accept [OK]

Common Mistakes:

• Thinking small p-value proves claim true
• Confusing p-value with probability of claim
• Assuming p-value alone proves claim false

2. Which of the following is the correct way to import the function for performing a t-test in SciPy?

easy

A. from scipy.stats import ttest_ind

B. import scipy.ttest as ttest_ind

C. from scipy import ttest_ind

D. import scipy.stats.ttest_ind()

Solution

1. Step 1: Recall SciPy stats module structure

   The t-test function is inside scipy.stats and is called ttest_ind.

2. Step 2: Check correct import syntax

   Correct syntax is 'from scipy.stats import ttest_ind' to import the function directly.

3. Final Answer:

   from scipy.stats import ttest_ind -> Option A

4. Quick Check:

   Correct import = from scipy.stats import ttest_ind [OK]

Hint: Use 'from scipy.stats import ttest_ind' to import t-test [OK]

Common Mistakes:

• Using incorrect module path
• Trying to import with parentheses
• Confusing import syntax with function call

3. Consider the following code using SciPy to test if two samples have different means:

import numpy as np
from scipy.stats import ttest_ind

sample1 = np.array([5, 7, 8, 6, 9])
sample2 = np.array([10, 12, 11, 13, 14])
stat, p = ttest_ind(sample1, sample2)
print(round(p, 3))

What is the printed p-value rounded to 3 decimals?

medium

A. 0.000

B. 0.050

C. 0.001

D. 0.100

Solution

1. Step 1: Run t-test on given samples

   Using ttest_ind on sample1 and sample2 calculates the p-value for difference in means.

2. Step 2: Calculate and round p-value

   Running the code gives p approximately 0.00002, which rounds to 0.000.

3. Final Answer:

   0.000 -> Option A

4. Quick Check:

   p-value ≈ 0.000 means very strong evidence against claim [OK]

Hint: Run code or estimate p-value from sample difference [OK]

Common Mistakes:

• Rounding p-value incorrectly
• Confusing test statistic with p-value
• Assuming p-value is zero without rounding

4. The following code attempts to perform a hypothesis test but has an error:

from scipy.stats import ttest_ind
sample1 = [1, 2, 3]
sample2 = [4, 5, 6]
stat, p = ttest_ind(sample1, sample2)
print(p)

What is the error and how to fix it?

medium

A. Import statement is incorrect, should be 'import scipy.stats.ttest_ind'

B. No error; code runs correctly

C. ttest_ind requires samples to be same length

D. sample1 and sample2 must be numpy arrays, convert with np.array()

Solution

1. Step 1: Check input types for ttest_ind

   ttest_ind accepts lists or arrays; lists are valid inputs.

2. Step 2: Verify code runs without error

   Running this code prints a p-value without error, so no fix needed.

3. Final Answer:

   No error; code runs correctly -> Option B

4. Quick Check:

   Lists accepted by ttest_ind = No error; code runs correctly [OK]

Hint: Lists work fine as input to ttest_ind [OK]

Common Mistakes:

• Thinking inputs must be numpy arrays
• Assuming sample sizes must match
• Incorrect import syntax

5. You want to test if a new teaching method improves test scores compared to the old method. You collect scores from two groups and run a t-test using SciPy. The p-value is 0.07. What is the best conclusion if you use a significance level of 0.05?

hard

A. The new method definitely does not improve scores

B. Accept the claim that the new method improves scores

C. Reject the claim that the new method improves scores

D. There is not enough evidence to reject the claim

Solution

1. Step 1: Compare p-value to significance level

   The p-value 0.07 is greater than 0.05, so we do not reject the null hypothesis.

2. Step 2: Interpret result in context

   Not rejecting means insufficient evidence to say the new method improves scores.

3. Final Answer:

   There is not enough evidence to reject the claim -> Option D

4. Quick Check:

   p > 0.05 means insufficient evidence [OK]

Hint: p > 0.05 means no strong evidence to reject claim [OK]

Common Mistakes:

• Confusing 'not reject' with 'accept claim'
• Thinking p=0.07 proves claim true
• Ignoring significance level threshold