Hypothesis testing (ttest, chi2test) in MATLAB - Mini Project: Build & Apply

Hypothesis Testing with t-test and Chi-square Test in MATLAB

📖 Scenario: You are a data analyst working with a small dataset. You want to check if the average height of a group is different from a known value, and also if two categorical variables are related.

🎯 Goal: Build a MATLAB script that performs a one-sample t-test and a chi-square test on given data.

📋 What You'll Learn

Create a vector of heights with exact values

Create a contingency table with exact counts

Use MATLAB's ttest function to test the mean height against a known value

Use MATLAB's chi2gof or chi2cont equivalent to test independence

Print the test results clearly

💡 Why This Matters

🌍 Real World

Hypothesis testing is used in many fields like medicine, marketing, and social sciences to make decisions based on data.

💼 Career

Data analysts and scientists use t-tests and chi-square tests to validate assumptions and find relationships in data.

1

Create the height data vector

Create a vector called heights with these exact values: [170, 172, 168, 165, 174, 169, 171].

MATLAB

% Create the vector heights with the given values
% Your code here

Hint

Use square brackets to create a vector in MATLAB.

2

Create the contingency table

Create a 2x2 matrix called contingencyTable with these exact counts: [30, 10; 20, 40].

MATLAB

heights = [170, 172, 168, 165, 174, 169, 171];
% Create the 2x2 matrix contingencyTable with the given counts
% Your code here

Hint

Use square brackets and semicolon to create a matrix in MATLAB.

3

Perform the one-sample t-test

Use MATLAB's ttest function to test if the mean of heights is different from 170. Store the test result in h_ttest and the p-value in p_ttest.

MATLAB

heights = [170, 172, 168, 165, 174, 169, 171];
contingencyTable = [30, 10; 20, 40];
% Perform the t-test comparing mean of heights to 170
% Your code here

Hint

Use [h, p] = ttest(data, mean_value) to perform a one-sample t-test.

4

Perform the chi-square test and display results

Use MATLAB's chi2gof function with the observed counts from contingencyTable to test independence. Store the test result in h_chi2 and the p-value in p_chi2. Then print the values of h_ttest, p_ttest, h_chi2, and p_chi2 with descriptive messages.

MATLAB

heights = [170, 172, 168, 165, 174, 169, 171];
contingencyTable = [30, 10; 20, 40];
[h_ttest, p_ttest] = ttest(heights, 170);
% Perform chi-square test on contingencyTable
% Print all test results with messages
% Your code here

Hint

Use chi2gof with observed and expected frequencies. Use fprintf to print results.

Practice

1. What does the MATLAB function ttest primarily test?

easy

A. Whether the data follows a normal distribution

B. Whether two samples have the same variance

C. Whether the mean of a sample differs from a specified value

D. Whether the counts in categories match expected counts

Solution

1. Step 1: Understand the purpose of ttest

   The ttest function in MATLAB tests if the mean of a sample is equal to a given value.

2. Step 2: Compare with other tests

   chi2gof tests counts, not means; variance tests are different; normality tests are separate.

3. Final Answer:

   Whether the mean of a sample differs from a specified value -> Option C

4. Quick Check:

   ttest tests mean difference [OK]

Hint: Remember: ttest checks means, chi2 tests counts [OK]

Common Mistakes:

• Confusing ttest with chi-square test
• Thinking ttest checks variance
• Assuming ttest checks distribution shape

2. Which of the following is the correct syntax to perform a one-sample t-test in MATLAB on data vector X against mean 0?

easy

A. chi2gof(X, 0)

B. ttest(X, 0)

C. ttest(0, X)

D. ttest(X)

Solution

1. Step 1: Recall ttest syntax

   The syntax ttest(X, mu) tests if mean of X equals mu.

2. Step 2: Check options

   chi2gof(X, 0) uses wrong function; ttest(0, X) reverses arguments; ttest(X) misses mean value.

3. Final Answer:

   ttest(X, 0) -> Option B

4. Quick Check:

   Correct syntax is ttest(data, mean) [OK]

Hint: Syntax: ttest(data, mean) for one-sample test [OK]

Common Mistakes:

• Swapping arguments order
• Using chi2gof for mean testing
• Omitting mean parameter

3. What is the output of the following MATLAB code?

X = [5.1, 5.3, 5.0, 5.2, 5.4];
[h, p] = ttest(X, 5.0);
disp(h); disp(p);

medium

A. h = 1, p < 0.05 (reject mean = 5.0)

B. h = 1, p > 0.05 (fail to reject mean = 5.0)

C. h = 0, p < 0.05 (reject mean = 5.0)

D. h = 0, p > 0.05 (fail to reject mean = 5.0)

Solution

1. Step 1: Compute sample statistics

   X mean = (5.1 + 5.3 + 5.0 + 5.2 + 5.4)/5 = 5.2; std ≈ 0.158.

2. Step 2: Interpret ttest output

   t ≈ (5.2 - 5.0) / (0.158 / √5) ≈ 2.83, df=4, p ≈ 0.047 < 0.05, so h=1 (reject null).

3. Final Answer:

   h = 1, p < 0.05 (reject mean = 5.0) -> Option A

4. Quick Check:

   mean=5.2, t≈2.83 > 2.776, p<0.05 [OK]

Hint: Sample mean 5.2 sufficiently differs from 5.0 (t>crit) -> h=1, p<0.05 [OK]

Common Mistakes:

• Assuming data too close to 5.0 for significance (h=0)
• Miscalculating p-value as >0.05
• Confusing h=1 (reject) meaning

4. Identify the error in this MATLAB code snippet for a chi-square goodness-of-fit test:

observed = [20, 30, 50];
expected = [25, 25, 50];
h = chi2gof(observed, 'ExpectedCounts', expected);

medium

A. The 'Expected' parameter is misspelled

B. The vectors observed and expected must be the same length

C. chi2gof cannot be used with expected counts

D. The observed data should be a vector of counts, but chi2gof expects raw data samples

Solution

1. Step 1: Understand chi2gof input requirements

   chi2gof expects raw sample data, not observed counts directly.

2. Step 2: Check usage of 'ExpectedCounts'

   The 'ExpectedCounts' parameter is valid but only used with raw data; providing counts directly leads to incorrect results.

3. Final Answer:

   The observed data should be a vector of counts, but chi2gof expects raw data samples -> Option D

4. Quick Check:

   chi2gof needs raw data, not counts [OK]

Hint: chi2gof needs raw data, not counts vector [OK]

Common Mistakes:

• Passing counts instead of raw data
• Misspelling parameter names
• Assuming chi2gof accepts counts directly

5. You have observed counts obs = [40, 35, 25] and expected proportions expProp = [0.3, 0.4, 0.3]. How do you correctly perform a chi-square goodness-of-fit test in MATLAB to check if observed counts fit expected proportions?

hard

A. Convert expected proportions to expected counts, then use chi2gof with raw data generated from counts

B. Directly use chi2gof(obs, 'Expected', expProp)

C. Use ttest(obs, expProp) to compare counts and proportions

D. Use chi2gof with observed counts and expected counts vectors

Solution

1. Step 1: Calculate expected counts from proportions

   Multiply total observed count by expected proportions: total = 40+35+25=100; expected counts = 100 * expProp = [30, 40, 30].

2. Step 2: Generate raw data samples for chi2gof

   Since chi2gof needs raw data, create a vector with repeated category labels matching observed counts.

3. Step 3: Perform chi-square test

   Use chi2gof with raw data and specify expected counts via 'ExpectedCounts' parameter.

4. Final Answer:

   Convert expected proportions to expected counts, then use chi2gof with raw data generated from counts -> Option A

5. Quick Check:

   chi2gof needs raw data and expected counts [OK]

Hint: Convert proportions to counts, create raw data for chi2gof [OK]

Common Mistakes:

• Passing proportions directly as expected counts
• Using ttest for count data
• Not converting counts to raw data samples