P-values and significance in Data Analysis Python - Time & Space Complexity

Time Complexity: P-values and significance

O(n)

Understanding Time Complexity

We want to understand how the time to calculate p-values changes as the amount of data grows.

How does the work needed grow when we have more data points?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

import numpy as np
from scipy import stats

def calculate_p_value(data1, data2):
    t_stat, p_val = stats.ttest_ind(data1, data2)
    return p_val

# data1 and data2 are lists or arrays of numbers

This code calculates the p-value from two groups of data using a t-test.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: The t-test function internally processes each data point in both groups.
• How many times: Each data point in both data1 and data2 is visited once to compute means and variances.

How Execution Grows With Input

As the number of data points increases, the time to compute the p-value grows roughly in direct proportion.

Input Size (n)	Approx. Operations
10	About 20 (10 in each group)
100	About 200
1000	About 2000

Pattern observation: Doubling the data roughly doubles the work needed.

Final Time Complexity

Time Complexity: O(n)

This means the time to calculate the p-value grows linearly with the total number of data points.

Common Mistake

[X] Wrong: "Calculating a p-value takes the same time no matter how much data there is."

[OK] Correct: The calculation must look at each data point to find averages and variances, so more data means more work.

Interview Connect

Understanding how data size affects calculation time helps you explain your approach clearly and shows you know what happens behind the scenes.

Self-Check

"What if we used a bootstrap method with 1000 resamples to estimate the p-value? How would the time complexity change?"

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