Why hypothesis testing validates claims in SciPy - Performance Analysis

Time Complexity: Why hypothesis testing validates claims

O(n)

Understanding Time Complexity

We want to see how the time needed to check a claim using hypothesis testing changes as we get more data.

How does the work grow when we test bigger datasets?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

import numpy as np
from scipy import stats

data = np.random.randn(n)  # generate n random data points
result = stats.ttest_1samp(data, popmean=0)  # test if mean is 0
p_value = result.pvalue
    

This code runs a t-test to check if the average of the data is different from zero.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Calculating the mean and variance of the data array.
• How many times: Each of the n data points is visited once to compute these statistics.

How Execution Grows With Input

As the number of data points grows, the time to calculate the test statistics grows roughly in direct proportion.

Input Size (n)	Approx. Operations
10	About 10 operations to sum and square values
100	About 100 operations
1000	About 1000 operations

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

Final Time Complexity

Time Complexity: O(n)

This means the time to validate a claim grows linearly with the number of data points.

Common Mistake

[X] Wrong: "Hypothesis testing time grows exponentially with data size because of complex calculations."

[OK] Correct: The main calculations just sum and average data once, so time grows steadily, not wildly.

Interview Connect

Understanding how hypothesis testing scales helps you explain data analysis steps clearly and shows you grasp practical data science skills.

Self-Check

"What if we used a bootstrap method with many resamples instead of a t-test? How would the time complexity change?"

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