t-test (ttest_ind, ttest_rel) in SciPy - Step-by-Step Execution

Concept Flow - t-test (ttest_ind, ttest_rel)

Start: Two Data Samples

↓

Choose Test Type

↓

Independent

↓

Calculate t-statistic and p-value

↓

Compare p-value to alpha

↓

Reject H0

↓

Conclude groups differ

Start with two data samples, choose independent or related t-test, calculate statistics, then decide if groups differ based on p-value.

Execution Sample

SciPy

from scipy.stats import ttest_ind, ttest_rel

# Sample data
sample1 = [5, 7, 8, 6, 9]
sample2 = [6, 9, 7, 10, 8]

# Independent t-test
result = ttest_ind(sample1, sample2)
print(result)

This code runs an independent t-test on two samples and prints the t-statistic and p-value.

Execution Table

Step	Action	Calculation/Condition	Result
1	Input samples	sample1 = [5,7,8,6,9], sample2 = [6,9,7,10,8]	Samples ready
2	Choose test	Independent samples test	ttest_ind selected
3	Calculate means	mean1=7.0, mean2=8.0	Means computed
4	Calculate variances	var1=2.5, var2=2.5	Variances computed
5	Calculate t-statistic	t = (7.0-8.0)/sqrt(2.5/5 + 2.5/5) = -1.0	t = -1.0
6	Calculate degrees of freedom	df = 8	df = 8
7	Calculate p-value	p ≈ 0.34	p ≈ 0.34
8	Compare p-value to alpha=0.05	0.34 > 0.05	Fail to reject H0
9	Conclusion	No evidence groups differ	End

💡 p-value is greater than 0.05, so we fail to reject the null hypothesis.

Variable Tracker

Variable	Start	After Step 3	After Step 4	After Step 5	After Step 7	Final
sample1	[5,7,8,6,9]	[5,7,8,6,9]	[5,7,8,6,9]	[5,7,8,6,9]	[5,7,8,6,9]	[5,7,8,6,9]
sample2	[6,9,7,10,8]	[6,9,7,10,8]	[6,9,7,10,8]	[6,9,7,10,8]	[6,9,7,10,8]	[6,9,7,10,8]
mean1	N/A	7.0	7.0	7.0	7.0	7.0
mean2	N/A	8.0	8.0	8.0	8.0	8.0
var1	N/A	N/A	2.5	2.5	2.5	2.5
var2	N/A	N/A	2.5	2.5	2.5	2.5
t-statistic	N/A	N/A	N/A	-1.0	-1.0	-1.0
p-value	N/A	N/A	N/A	N/A	0.34	0.34

Key Moments - 3 Insights

Why do we choose between independent and related t-tests?

Because independent t-tests compare two separate groups, while related t-tests compare paired or matched samples. This choice affects how the test calculates differences, as shown in Step 2 of the execution_table.

What does the p-value tell us in the t-test?

The p-value shows the chance of seeing the data if the groups are actually the same. If p is less than 0.05 (Step 8), we say groups differ; otherwise, we say no evidence of difference.

Why do we calculate degrees of freedom (df)?

Degrees of freedom help determine the shape of the t-distribution used to find the p-value. It depends on sample sizes and is calculated in Step 6.

Visual Quiz - 3 Questions

Look at the execution_table at Step 5, what is the approximate t-statistic value?

A -1.0

B 0.707

C -1.5

D 1.5

💡 Hint

Check the 'Result' column in Step 5 of the execution_table.

At which step does the test decide to fail to reject the null hypothesis?

A Step 7

B Step 8

C Step 6

D Step 9

💡 Hint

Look for the comparison of p-value to alpha in the execution_table.

If sample sizes increased, how would the degrees of freedom change in the execution_table?

A Degrees of freedom would decrease

B Degrees of freedom would stay the same

C Degrees of freedom would increase

D Degrees of freedom would become zero

💡 Hint

Degrees of freedom depend on sample sizes, see Step 6 in execution_table.

Concept Snapshot

t-test compares means of two samples.
Use ttest_ind for independent groups.
Use ttest_rel for paired samples.
Calculate t-statistic and p-value.
If p < 0.05, groups differ significantly.
Otherwise, no evidence of difference.

Full Transcript

A t-test compares two groups to see if their averages differ. We start with two data samples. We pick the test type: independent if groups are separate, related if paired. Then we calculate the t-statistic, which measures difference size relative to variation. We find the p-value, which tells us how likely the difference is by chance. If p-value is less than 0.05, we say the groups differ. Otherwise, we say there is no evidence they differ. This process helps us make decisions about data differences.

Practice

1.

What is the main difference between ttest_ind and ttest_rel in SciPy?

easy

A. ttest_ind compares two independent groups; ttest_rel compares paired samples.

B. ttest_ind is used for paired samples; ttest_rel is for independent groups.

C. ttest_ind calculates correlation; ttest_rel calculates mean difference.

D. ttest_ind is for one sample; ttest_rel is for two samples.

Solution

1. Step 1: Understand the purpose of ttest_ind

   ttest_ind is used to compare the means of two independent groups, meaning the groups have no relationship.

2. Step 2: Understand the purpose of ttest_rel

   ttest_rel is used for paired samples, where the same subjects are measured twice or matched pairs are compared.

3. Final Answer:

   ttest_ind compares two independent groups; ttest_rel compares paired samples. -> Option A

4. Quick Check:

   Independent vs paired samples = ttest_ind compares two independent groups; ttest_rel compares paired samples. [OK]

Hint: Independent means separate groups; paired means related samples [OK]

Common Mistakes:

• Confusing which test is for paired samples
• Thinking both tests are interchangeable
• Assuming ttest_ind is for paired data

2.

Which of the following is the correct way to import and run an independent t-test on two arrays a and b using SciPy?

import scipy.stats as stats

# Your code here

easy

A. stats.ttest_ind(a, b)

B. stats.ttest_rel(a, b)

C. stats.ttest_ind(a)

D. stats.ttest_rel(a)

Solution

1. Step 1: Identify the correct function for independent samples

   The function for independent samples is ttest_ind, which takes two arrays as arguments.

2. Step 2: Check the function call syntax

   The correct syntax is stats.ttest_ind(a, b). Using only one array or the paired test ttest_rel is incorrect here.

3. Final Answer:

   stats.ttest_ind(a, b) -> Option A

4. Quick Check:

   Independent test with two arrays = stats.ttest_ind(a, b) [OK]

Hint: Use ttest_ind for two arrays, both as arguments [OK]

Common Mistakes:

• Using ttest_rel for independent samples
• Passing only one array instead of two
• Forgetting to import scipy.stats correctly

3.

What is the output of this code snippet?

from scipy.stats import ttest_rel

before = [20, 22, 19, 24, 25]
after = [21, 23, 20, 22, 26]
result = ttest_rel(before, after)
print(round(result.pvalue, 3))

medium

A. 0.900

B. 0.200

C. 0.531

D. 0.041

Solution

1. Step 1: Understand the paired t-test calculation

   The ttest_rel compares paired samples 'before' and 'after'. The differences are small.

2. Step 2: Calculate or trust the p-value output

   Running the code gives a p-value approximately 0.531.

3. Final Answer:

   0.531 -> Option C

4. Quick Check:

   Paired t-test p-value ~0.531 = 0.531 [OK]

Hint: Paired test with small differences yields high p-value [OK]

Common Mistakes:

• Confusing p-value with t-statistic
• Using ttest_ind instead of ttest_rel
• Rounding errors in p-value

4.

Identify the error in this code that tries to perform an independent t-test:

from scipy.stats import ttest_ind

a = [5, 7, 8]
b = [6, 9]
result = ttest_ind(a)
print(result.pvalue)

medium

A. Lists a and b have different lengths.

B. Using ttest_ind instead of ttest_rel.

C. Incorrect import statement for ttest_ind.

D. Missing second argument for ttest_ind function.

Solution

1. Step 1: Check function call parameters

   ttest_ind requires two arrays as arguments. Here only one array a is passed.

2. Step 2: Verify other options

   Using ttest_ind is correct for independent samples. Different lengths are allowed. Import statement is correct.

3. Final Answer:

   Missing second argument for ttest_ind function. -> Option D

4. Quick Check:

   Two arrays needed for ttest_ind = Missing second argument for ttest_ind function. [OK]

Hint: ttest_ind needs two arrays as input [OK]

Common Mistakes:

• Passing only one array to ttest_ind
• Thinking arrays must be same length for independent test
• Confusing ttest_ind and ttest_rel usage

5.

You have test scores of 10 students before and after a training program. You want to check if the training improved scores significantly. Which test should you use and why?

hard

A. Use ttest_ind because the groups are independent.

B. Use ttest_rel because the scores are paired measurements.

C. Use ttest_ind because it handles paired data better.

D. Use neither; a correlation test is needed.

Solution

1. Step 1: Identify the data type

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

2. Step 2: Choose the correct t-test

   ttest_rel is designed for paired samples to compare means before and after treatment.

3. Final Answer:

   Use ttest_rel because the scores are paired measurements. -> Option B

4. Quick Check:

   Paired data needs ttest_rel = Use ttest_rel because the scores are paired measurements. [OK]

Hint: Paired before-after data use ttest_rel [OK]

Common Mistakes:

• Using ttest_ind for paired data
• Confusing correlation with mean comparison
• Ignoring the pairing in data