Why hypothesis testing validates claims in SciPy - Challenge Your Understanding

Challenge - 5 Problems

🧠 Conceptual

intermediate

What does the p-value represent in hypothesis testing?

In hypothesis testing, what does the p-value tell us about the claim we are testing?

A The probability that the observed data is incorrect.

B The probability that the alternative hypothesis is true given the data.

C The probability that the null hypothesis is false.

D The probability of observing the data, or something more extreme, assuming the null hypothesis is true.

💡 Hint

Think about what the p-value measures under the assumption that the null hypothesis holds.

The p-value measures how likely it is to get the observed data (or more extreme) if the null hypothesis is true. It does not directly tell us the probability that the null or alternative hypothesis is true.

❓ Predict Output

intermediate

Output of a t-test on sample data

Given the following Python code using scipy.stats, what is the output of the t-statistic and p-value?

SciPy

from scipy import stats

sample = [5, 7, 8, 6, 9, 7, 5]
# Test if the sample mean is different from 6
result = stats.ttest_1samp(sample, 6)
print(round(result.statistic, 2), round(result.pvalue, 3))

A 1.26 0.250

B 0.87 0.412

C 2.45 0.045

D 3.12 0.015

💡 Hint

Calculate the sample mean and standard deviation, then apply the t-test formula or run the code.

The t-test compares the sample mean to the population mean 6. The calculated t-statistic is about 1.26 and the p-value about 0.250, meaning no strong evidence to reject the null hypothesis.

❓ data_output

advanced

Result of a two-sample t-test on two groups

Two groups have the following data:

Group A: [12, 15, 14, 10, 13]

Group B: [8, 9, 7, 6, 10]

Using a two-sample t-test, what is the t-statistic and p-value?

SciPy

from scipy import stats

group_a = [12, 15, 14, 10, 13]
group_b = [8, 9, 7, 6, 10]
result = stats.ttest_ind(group_a, group_b)
print(round(result.statistic, 2), round(result.pvalue, 3))

A 4.31 0.002

B 2.13 0.075

C 3.58 0.010

D 1.98 0.120

💡 Hint

Calculate the means and variances of both groups, then apply the two-sample t-test or run the code.

The t-statistic is about 4.31 with a p-value of 0.002, indicating strong evidence that the two groups have different means.

❓ visualization

advanced

Interpreting a p-value distribution plot

Below is a plot showing the distribution of p-values from many hypothesis tests under the null hypothesis. What does the uniform distribution of p-values indicate?

SciPy

import matplotlib.pyplot as plt
import numpy as np

np.random.seed(0)
p_values = np.random.uniform(0, 1, 1000)
plt.hist(p_values, bins=20, edgecolor='black')
plt.title('Distribution of p-values under Null Hypothesis')
plt.xlabel('p-value')
plt.ylabel('Frequency')
plt.show()

A Most p-values cluster near zero indicating strong evidence against the null hypothesis.

B P-values are equally likely across the range 0 to 1 when the null hypothesis is true.

C P-values cluster near one indicating the null hypothesis is always true.

D P-values follow a normal distribution centered at 0.5.

💡 Hint

Think about what the p-value distribution looks like if the null hypothesis is true for all tests.

When the null hypothesis is true, p-values are uniformly distributed between 0 and 1, meaning any p-value is equally likely.

🔧 Debug

expert

Identify the error in hypothesis testing code

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

from scipy import stats
sample = [2, 4, 6, 8]
result = stats.ttest_1samp(sample)
print(result)

A No error, prints t-statistic and p-value

B ValueError: Sample size must be greater than 1

C TypeError: ttest_1samp() missing 1 required positional argument: 'popmean'

D NameError: name 'stats' is not defined

💡 Hint

Check the required arguments for ttest_1samp function.

The function ttest_1samp requires two arguments: the sample and the population mean to test against. Omitting the population mean causes a TypeError.