One-Sample t-Test Using Python

Example 1 — Test Anxiety Score

1. State the Research Problem or Question

Research question: Is the mean test-anxiety score of the students significantly different from the reference score of 8?

This investigates whether the average test-anxiety score observed in one group of students differs from a specified benchmark.

2. Identify the Variables

3. State the Hypotheses

H₀: μ = 8

H₁: μ ≠ 8

This is a two-tailed test because the question asks whether the mean is different from 8.

4. Set the Level of Significance

α = 0.05

5. Explain Why the Statistical Test Is Appropriate

• Number of groups/conditions: One sample is compared with one reference mean.
• Independent or related observations: Observations are independent because each score belongs to a different student.
• Measurement level: Test-anxiety score is treated as a continuous variable.
• Important assumptions: No significant outliers are observed, and the distribution is approximately normal.
• Normality evidence: Shapiro–Wilk W = 0.961, p = 0.560, so there is no significant evidence of non-normality.

6. Present the Python Results

Only the important output is presented. Lengthy manual computations are not required.

7. Make the Statistical Decision

Because p = 0.0009 < α = 0.05, reject H₀.

8. Interpret the Result

Example 2 — Programming Examination Score

1. State the Research Problem or Question

Research question: Is the mean programming examination score of the students significantly different from 85?

This investigates whether the average score of one group of BSIT students differs from the expected benchmark score of 85.

2. Identify the Variables

3. State the Hypotheses

H₀: μ = 85

H₁: μ ≠ 85

4. Set the Level of Significance

α = 0.05

5. Explain Why the Statistical Test Is Appropriate

• Number of groups/conditions: One sample is compared with a reference mean of 85.
• Independent or related observations: Each score comes from a different student, so the observations are independent.
• Measurement level: Examination score is treated as continuous.
• Important assumptions: No serious outliers are observed, and the scores are approximately normally distributed.
• Normality evidence: Shapiro–Wilk W = 0.965, p = 0.419.

6. Present the Python Results

7. Make the Statistical Decision

Because p < α = 0.05, reject H₀.

8. Interpret the Result

Example 3 — Information-System Response Time

1. State the Research Problem or Question

Research question: Is the average system response time significantly different from 250 milliseconds?

This investigates whether the observed average response time of the system differs from the specified performance benchmark.

2. Identify the Variables

3. State the Hypotheses

H₀: μ = 250 ms

H₁: μ ≠ 250 ms

4. Set the Level of Significance

α = 0.05

5. Explain Why the Statistical Test Is Appropriate

• Number of groups/conditions: One set of response-time observations is compared with a benchmark of 250 ms.
• Independent or related observations: Each recorded response is treated as an independent observation.
• Measurement level: Response time measured in milliseconds is continuous.
• Important assumptions: No significant outliers are observed, and the response times are approximately normal.
• Normality evidence: Shapiro–Wilk W = 0.979, p = 0.872.

6. Present the Python Results

7. Make the Statistical Decision

Because p < α = 0.05, reject H₀.

8. Interpret the Result

Suggested Python Code

import pandas as pd
from scipy.stats import ttest_1samp, shapiro

data = pd.read_csv("data.csv")
scores = data["Score"]

print("Mean:", scores.mean())
print("Standard deviation:", scores.std())
print("Sample size:", len(scores))

w, p_normal = shapiro(scores)
print("Shapiro-Wilk W:", w)
print("Normality p-value:", p_normal)

t_stat, p_value = ttest_1samp(scores, popmean=85)

print("t-statistic:", t_stat)
print("p-value:", p_value)

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