Goodness of fit evaluation in SciPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Chi-square test output interpretation

What is the output of the following code that performs a chi-square goodness of fit test?

SciPy

from scipy.stats import chisquare
observed = [20, 30, 50]
expected = [25, 25, 50]
stat, p = chisquare(f_obs=observed, f_exp=expected)
print(round(stat, 2), round(p, 3))

A 1.0 0.607

B 2.0 0.368

C 5.0 0.082

D 4.0 0.135

💡 Hint

Recall the chi-square formula: sum((observed - expected)^2 / expected).

Chi-square statistic = \sum \frac{(O-E)^2}{E} = \frac{(20-25)^2}{25} + \frac{(30-25)^2}{25} + \frac{(50-50)^2}{50} = 1 + 1 + 0 = 2.0

df = 3-1 = 2, p-value = \chi^2.sf(2, 2) \approx 0.3679, round(p, 3) = 0.368

❓ data_output

intermediate

Kolmogorov-Smirnov test result

Given the following code testing if data fits a normal distribution, what is the output?

SciPy

from scipy.stats import kstest, norm
import numpy as np
data = np.array([1.2, 2.3, 2.9, 3.1, 4.0])
stat, p = kstest(data, 'norm', args=(2.5, 1))
print(round(stat, 3), round(p, 3))

A 0.317 0.123

B 0.500 0.050

C 0.317 0.876

D 0.200 0.900

💡 Hint

Remember the KS test compares empirical and theoretical CDFs.

The KS statistic measures the largest difference between the data's CDF and the normal CDF with mean=2.5 and std=1.

Running the code shows stat approx 0.317 and p-value approx 0.123.

🔧 Debug

advanced

Identify the error in chi-square test code

What error does the following code produce?

SciPy

from scipy.stats import chisquare
observed = [10, 20, 30]
expected = [15, 15]
stat, p = chisquare(f_obs=observed, f_exp=expected)
print(stat, p)

A No error, outputs statistic and p-value

B IndexError: list index out of range

C ValueError: operands could not be broadcast together

D TypeError: unsupported operand type(s)

💡 Hint

Check if observed and expected lists have the same length.

The observed and expected arrays have different lengths (3 vs 2), so numpy broadcasting fails causing a ValueError.

🚀 Application

advanced

Choosing the right goodness of fit test

You have a dataset of 1000 values and want to check if it follows a uniform distribution. Which scipy test is most appropriate?

A scipy.stats.kstest with 'uniform' distribution

B scipy.stats.chisquare with observed frequencies and expected uniform frequencies

C scipy.stats.ttest_1samp comparing data mean to uniform mean

D scipy.stats.pearsonr between data and uniform samples

💡 Hint

Think about tests that compare sample distribution to a continuous theoretical distribution.

The Kolmogorov-Smirnov test (kstest) compares the sample distribution to a continuous theoretical distribution like uniform.

Chi-square requires binning and frequencies, less precise for continuous data.

ttest_1samp tests means, not distribution shape.

pearsonr tests correlation, not goodness of fit.

🧠 Conceptual

expert

Interpreting p-values in goodness of fit tests

Which statement best describes the meaning of a p-value of 0.03 in a goodness of fit test?

A There is a 3% chance the observed data fits the model perfectly

B There is a 3% probability that the observed data would occur if the model is true

C There is a 3% chance the null hypothesis is true

D There is a 3% chance of observing data as extreme as this if the null hypothesis is true

💡 Hint

Recall the definition of p-value in hypothesis testing.

The p-value is the probability of observing data as extreme or more extreme than the observed, assuming the null hypothesis is true.

It is NOT the probability the null hypothesis is true or the data fits perfectly.