Why statistics validates hypotheses in Data Analysis Python - Challenge Your Understanding

Challenge - 5 Problems

🧠 Conceptual

intermediate

Understanding Hypothesis Testing Purpose

Why do we use statistics to validate hypotheses in data science?

A To randomly select data points for analysis

B To ignore data variability and focus only on averages

C To measure the likelihood that observed data could happen if the hypothesis is false

D To prove the hypothesis is absolutely true without any doubt

💡 Hint

Think about what statistics tells us about data and hypotheses.

Statistics helps us understand how likely our observed data would be if the hypothesis were not true. It does not prove absolute truth but measures evidence strength.

❓ Predict Output

intermediate

Output of p-value calculation

What is the output of this Python code that calculates a p-value for a sample mean?

Data Analysis Python

import scipy.stats as stats
sample_mean = 105
population_mean = 100
std_dev = 15
sample_size = 30
z_score = (sample_mean - population_mean) / (std_dev / (sample_size ** 0.5))
p_value = 2 * (1 - stats.norm.cdf(abs(z_score)))
print(round(p_value, 4))

A 0.0910

B 0.0455

C 0.9545

D 0.1820

💡 Hint

Calculate z-score first, then find two-tailed p-value.

The z-score is about 1.8257, so the two-tailed p-value is approximately 0.0455, indicating significance at 5% level.

❓ data_output

advanced

Result of Hypothesis Test on Sample Data

Given this sample data and a population mean of 50, what is the p-value from a one-sample t-test?

Data Analysis Python

import scipy.stats as stats
sample = [52, 48, 51, 49, 53, 47, 50, 54, 46, 55]
population_mean = 50
t_stat, p_value = stats.ttest_1samp(sample, population_mean)
print(round(p_value, 4))

A 0.0543

B 0.1234

C 0.9876

D 0.0027

💡 Hint

Use the t-test formula for one sample against population mean.

The t-test compares sample mean to population mean; the p-value of 0.0543 suggests borderline significance.

🔧 Debug

advanced

Identify the error in hypothesis test code

What error will this code raise when running a t-test?

Data Analysis Python

import scipy.stats as stats
sample = [5, 7, 8, 6, 9]
pop_mean = 7
result = stats.ttest_1samp(sample, pop_mean)
print(result.pvalue)

A AttributeError: 'tuple' object has no attribute 'pvalue'

B SyntaxError: invalid syntax

C TypeError: ttest_1samp() missing 1 required positional argument

D No error, prints the p-value

💡 Hint

Check what ttest_1samp returns and how to access p-value.

stats.ttest_1samp returns a tuple (statistic, pvalue), so accessing result.pvalue causes AttributeError.

🚀 Application

expert

Choosing correct hypothesis test for data

You have two independent groups with small sample sizes and unknown population variances. Which test should you use to validate if their means differ?

A Chi-square test

B Two-sample t-test assuming equal variances

C Paired t-test

D Two-sample t-test with Welch's correction

💡 Hint

Consider variance equality and sample independence.

Welch's t-test does not assume equal variances and is suitable for small samples with unknown variances.

Practice

1. What does a small p-value indicate in hypothesis testing?

easy

A. The data is random and unimportant

B. Strong evidence against the null hypothesis

C. The hypothesis is definitely true

D. The sample size is too small

Solution

1. Step 1: Understand p-value meaning

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

2. Step 2: Interpret small p-value

   A small p-value means the observed data is unlikely under the null hypothesis, so we have strong evidence against it.

3. Final Answer:

   Strong evidence against the null hypothesis -> Option B

4. Quick Check:

   Small p-value = reject null hypothesis [OK]

Hint: Small p-value means reject null hypothesis [OK]

Common Mistakes:

• Thinking small p-value proves hypothesis true
• Confusing p-value with sample size
• Assuming p-value shows data randomness

2. Which Python library is commonly used to perform hypothesis testing?

easy

A. matplotlib

B. seaborn

C. scipy.stats

D. pandas

Solution

1. Step 1: Identify libraries for statistics

   matplotlib and seaborn are for plotting; pandas is for data handling.

2. Step 2: Recognize scipy.stats role

   scipy.stats provides functions for statistical tests including hypothesis testing.

3. Final Answer:

   scipy.stats -> Option C

4. Quick Check:

   scipy.stats = stats tests [OK]

Hint: Use scipy.stats for hypothesis tests in Python [OK]

Common Mistakes:

• Choosing plotting libraries for testing
• Confusing data handling with testing
• Not knowing scipy.stats functions

3. What is the output of this Python code snippet?

from scipy import stats
sample = [5, 7, 8, 6, 9]
result = stats.ttest_1samp(sample, 7)
print(round(result.pvalue, 3))

medium

A. 0.317

B. 0.045

C. 0.500

D. 0.001

Solution

1. Step 1: Understand t-test function

   stats.ttest_1samp tests if sample mean differs from 7.

2. Step 2: Calculate p-value

   Running the code gives p-value approximately 0.317, meaning no strong evidence against mean=7.

3. Final Answer:

   0.317 -> Option A

4. Quick Check:

   ttest p-value ≈ 0.317 [OK]

Hint: Run code to get exact p-value [OK]

Common Mistakes:

• Assuming p-value is always below 0.05
• Mixing up test statistic and p-value
• Rounding errors in p-value

4. Identify the error in this hypothesis test code:

from scipy import stats
sample = [10, 12, 9, 11]
p_value = stats.ttest_1samp(sample, 10)
if p_value < 0.05:
    print("Reject null hypothesis")
else:
    print("Fail to reject null hypothesis")

medium

A. p_value is a tuple, should use p_value.pvalue

B. Comparison operator should be == not <

C. Sample size is too small for t-test

D. Missing import for numpy

Solution

1. Step 1: Check return type of ttest_1samp

   stats.ttest_1samp returns an object with statistic and pvalue, not a single number.

2. Step 2: Correct p-value access

   We must use p_value.pvalue to get the p-value number for comparison.

3. Final Answer:

   p_value is a tuple, should use p_value.pvalue -> Option A

4. Quick Check:

   Use .pvalue attribute to get p-value [OK]

Hint: Use .pvalue attribute from ttest result [OK]

Common Mistakes:

• Comparing whole result object directly
• Wrong comparison operator
• Assuming numpy import needed here

5. You want to test if a new teaching method improves scores. You collect scores from 30 students before and after. Which statistical test best validates your hypothesis?

hard

A. Independent two-sample t-test

B. Chi-square test

C. One-sample z-test

D. Paired t-test

Solution

1. Step 1: Identify data type and design

   Scores are from same students before and after, so data is paired.

2. Step 2: Choose correct test for paired data

   Paired t-test compares means of related samples to check improvement.

3. Final Answer:

   Paired t-test -> Option D

4. Quick Check:

   Paired data needs paired t-test [OK]

Hint: Use paired t-test for before-after same subjects [OK]

Common Mistakes:

• Using independent t-test for paired data
• Choosing chi-square for numeric scores
• Using z-test without population variance