P-values and significance in Data Analysis Python

Introduction

P-values help us decide if a result is likely due to chance or if it shows a real effect.

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

Testing if a marketing campaign increased sales.

Seeing if students' test scores improved after extra tutoring.

Determining if a coin is fair or biased.

Syntax

Data Analysis Python

from scipy import stats

# Calculate p-value from test statistic and degrees of freedom
p_value = stats.t.sf(abs(t_stat), df)*2

# Or directly from data samples
stat, p_value = stats.ttest_ind(sample1, sample2)

P-value is a number between 0 and 1.

Lower p-values mean stronger evidence against chance.

Examples

Tests if sample mean is different from 6.

Data Analysis Python

from scipy import stats

# Example 1: One sample t-test
sample = [5, 6, 7, 8, 9]
stat, p_value = stats.ttest_1samp(sample, 6)
print(f"p-value: {p_value:.3f}")

Tests if two samples have different means.

Data Analysis Python

from scipy import stats

# Example 2: Two sample t-test
sample1 = [5, 6, 7, 8, 9]
sample2 = [7, 8, 9, 10, 11]
stat, p_value = stats.ttest_ind(sample1, sample2)
print(f"p-value: {p_value:.3f}")

Sample Program

This code compares two groups' exam scores to check if they differ significantly.

Data Analysis Python

from scipy import stats

# Two groups of exam scores
group_A = [85, 87, 90, 88, 86]
group_B = [80, 82, 79, 81, 83]

# Perform t-test to see if scores differ
stat, p_value = stats.ttest_ind(group_A, group_B)

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

# Decide significance at 0.05 level
if p_value < 0.05:
    print("Result is significant: groups differ.")
else:
    print("Result is not significant: no clear difference.")

Important Notes

A common cutoff for significance is 0.05, meaning 5% chance results are random.

P-values do not measure size of effect, only evidence against chance.

Always consider context and data quality, not just p-value.

Summary

P-values help decide if results are likely real or by chance.

Lower p-values mean stronger evidence against randomness.

Use p-values with a significance level like 0.05 to make decisions.

Practice

1. What does a p-value less than 0.05 usually indicate in a hypothesis test?

easy

A. The results are statistically significant and unlikely due to chance.

B. The results are definitely caused by random chance.

C. The sample size is too small to draw conclusions.

D. The test was performed incorrectly.

Solution

1. Step 1: Understand the meaning of p-value

   A p-value measures how likely the observed data would occur if the null hypothesis were true (random chance).

2. Step 2: Compare p-value to significance level

   If the p-value is less than 0.05 (common threshold), it means the observed result is unlikely due to chance, so we reject the null hypothesis.

3. Final Answer:

   The results are statistically significant and unlikely due to chance. -> Option A

4. Quick Check:

   p-value < 0.05 means significant [OK]

Hint: p-value < 0.05 means strong evidence against chance [OK]

Common Mistakes:

• Thinking p-value shows probability null is true
• Confusing significance level with p-value
• Assuming p-value proves practical importance

2. Which of the following is the correct way to check if a p-value stored in variable p_val is significant at 0.05 level in Python?

easy

A. if p_val == 0.05: print('Significant')

B. if p_val > 0.05: print('Significant')

C. if p_val < 0.05: print('Significant')

D. if p_val <= 0.5: print('Significant')

Solution

1. Step 1: Recall significance condition

   A p-value less than 0.05 means significance, so we check if p_val < 0.05.

2. Step 2: Match condition to code

   if p_val < 0.05: print('Significant') uses < 0.05 correctly. Others check wrong conditions or wrong thresholds.

3. Final Answer:

   if p_val < 0.05: print('Significant') -> Option C

4. Quick Check:

   Use < 0.05 for significance [OK]

Hint: Check if p_val < 0.05 for significance [OK]

Common Mistakes:

• Using > instead of < for significance
• Checking equality to 0.05 exactly
• Using wrong threshold like 0.5

3. Given this Python code snippet, what will be printed?

p_val = 0.07
if p_val < 0.05:
    print('Reject null hypothesis')
else:
    print('Fail to reject null hypothesis')

medium

A. Reject null hypothesis

B. SyntaxError

C. No output

D. Fail to reject null hypothesis

Solution

1. Step 1: Check the p-value against 0.05

   The p_val is 0.07, which is greater than 0.05, so the condition p_val < 0.05 is False.

2. Step 2: Determine which branch runs

   Since the condition is False, the else block runs, printing 'Fail to reject null hypothesis'.

3. Final Answer:

   Fail to reject null hypothesis -> Option D

4. Quick Check:

   p_val = 0.07 > 0.05 means fail to reject [OK]

Hint: p_val > 0.05 means fail to reject null [OK]

Common Mistakes:

• Confusing greater than with less than
• Assuming equality means reject
• Ignoring else branch

4. This code is intended to print 'Significant' if p-value is less than 0.05, but it always prints 'Not Significant'. What is the error?

p_val = 0.03
if p_val > 0.05:
    print('Significant')
else:
    print('Not Significant')

medium

A. The comparison operator should be '<' instead of '>'

B. The variable p_val is not defined

C. The print statements are reversed

D. The p_val value is incorrect

Solution

1. Step 1: Analyze the condition logic

   The code checks if p_val > 0.05 to print 'Significant', but significance means p_val < 0.05.

2. Step 2: Identify the fix

   Changing the operator to < will correctly print 'Significant' when p_val is 0.03.

3. Final Answer:

   The comparison operator should be '<' instead of '>' -> Option A

4. Quick Check:

   Use < 0.05 for significance check [OK]

Hint: Use '<' to check p-value significance [OK]

Common Mistakes:

• Using '>' instead of '<' for significance
• Swapping print statements without fixing condition
• Assuming variable is undefined

5. You run a test and get a p-value of 0.049. You want to be very strict and use a significance level of 0.01. What should you conclude?

hard

A. Reject the null hypothesis because 0.049 < 0.05

B. Fail to reject the null hypothesis because 0.049 > 0.01

C. Reject the null hypothesis because 0.049 < 0.01

D. Fail to reject the null hypothesis because 0.049 > 0.05

Solution

1. Step 1: Understand significance level

   The significance level is set to 0.01, meaning p-values must be less than 0.01 to reject the null hypothesis.

2. Step 2: Compare p-value to significance level

   The p-value 0.049 is greater than 0.01, so we fail to reject the null hypothesis at this stricter level.

3. Final Answer:

   Fail to reject the null hypothesis because 0.049 > 0.01 -> Option B

4. Quick Check:

   p-value > significance level means fail to reject [OK]

Hint: Compare p-value to chosen significance level carefully [OK]

Common Mistakes:

• Using default 0.05 without checking chosen level
• Confusing less than and greater than
• Ignoring stricter significance level