😴 Sleepothesis : Does Sleep Improve Academic Performance?

Introduction :

We always hear:

“You must sleep 8 hours a day.”

But is that scientifically true for everyone?

For months, I kept asking myself:

“Do I actually score better when I sleep 8 hours… or is it just random?”

Some days I slept 5 hours and struggled through mock tests. Other days I slept 8 hours and felt unstoppable.

Instead of guessing, I decided to test it statistically.

And not just with one method.

Case Study :

For 30 days, I recorded:

• Hours of sleep
• Mock test score
• Study method
• Productivity level
• Pass/Fail

Goal:

Does sleeping 8 hours improve my exam performance?

This is where Hypothesis Testing helps.

What Is Hypothesis Testing?

Hypothesis testing is a statistical method used to determine whether there is enough evidence to support a claim.

“Is this effect real, or is it just random variation?”

Types of Hypothesis Testing :

PARAMETRIC TESTS

Parametric tests assume:

• Data is numerical
• Data is normally distributed
• We are testing population means

In my case:

• My mock test scores are numerical
• I assume approximate normal distribution

So parametric tests apply.

1.1 Z-Test

What is Z-Test?

A Z-test is used to compare a sample mean with a known population mean when:

• Population standard deviation (σ) is known
• Sample size is large (n ≥ 30)

Last year (before fixing sleep):

• My average score = 70
• Population standard deviation = 10

After following 8-hour sleep routine for 30 days:

• Sample mean = 75
• n = 30

Now we test different types of claims.

Two-Tailed Z-Test

Question:

Is my new average score different from my old average of 70?

Given:

Old mean (μ) = 70 Population SD (σ) = 10 New sample mean (x̄) = 75 Sample size (n) = 30

Hypotheses

Reject H₀

Conclusion:

My score is significantly different after following the 8-hour sleep routine.

Interpretation

P-value ≈ 0.006

Since:

0.006 < 0.05

Reject Null Hypothesis

Special Case Where Z-Test Fails

Z-test becomes invalid when:

1. Population SD Unknown

If I don’t know σ = 10 → Must use T-test instead.

2. Small Sample Size

If I only tracked sleep for 5 days → Z-test assumption breaks.

3. Non-Normal Data

If my scores are extremely skewed → Z-test may give misleading result.

4. Outliers

One extreme score can distort mean heavily.

Right-Tailed Z-Test

A right-tailed Z-test checks whether the sample mean is greater than the population mean.

It is used when your claim is about improvement or increase.

After sleeping 8 hours for 30 days:

Did my average score increase above 70?

Hypotheses :

Given Data

Population mean (μ) = 70 Population SD (σ) = 10 Sample mean (x̄) = 75 Sample size (n) = 30

Decision Rule

Reject H₀

Left-Tailed Z-Test

A left-tailed Z-test checks whether the sample mean is less than the population mean.

Used when testing for decline or reduction.

What if I suspect:

Sleeping more made me less productive?

Hypotheses

Suppose New Sample Mean = 66

Decision Rule

Reject H₀

What is a T-Test?

A T-test is used to determine whether a sample mean is significantly different from a population mean (or another sample mean) when the population standard deviation is unknown.

It uses the t-distribution, which has heavier tails than the normal distribution.

After following my 8-hour sleep routine, I collected data for 10 days.

Old average score (before fixing sleep): μ = 70

New 10-day scores after sleeping 8 hours:

72, 75, 78, 74, 76, 73, 77, 79, 71, 75

We now test whether my new mean differs from 70.

Sample size n = 10 Degrees of freedom df = 9

Two-Tailed T-Test

Question:

Is my new average score different from 70?

Hypotheses:

H₀: μ = 70

H₁: μ ≠ 70

The regions after [-6,6] represent extreme outcomes under the null hypothesis.

Right-Tailed T-Test

Question:

Did sleeping 8 hours improve my score?

Hypotheses:

H₀: μ = 70 H₁: μ > 70

Left-Tailed T-Test

Now suppose my new scores were lower.

Example new scores:

65, 67, 66, 64, 68, 63, 69, 66, 67, 65

Sample mean ≈ 66

Hypotheses:

H₀: μ = 70 H₁: μ < 70

If p_left < 0.05 Reject H₀.

When T-Test Fails

The T-test becomes unreliable when:

• Data is extremely skewed
• Strong outliers exist
• Sample size is very small and non-normal
• Data is categorical

In those cases, we move to non-parametric tests such as Chi-Square.

What is ANOVA?

ANOVA (Analysis of Variance) is used to determine whether there are statistically significant differences between the means of three or more independent groups.

Instead of comparing means directly, ANOVA compares variance between groups and variance within groups.

Why Not Use Multiple T-Tests?

If we perform many t-tests:

• Type I error increases
• Results become unreliable

ANOVA controls this error using a single test.

ANOVA Formula :

Now instead of just 8 hours, I test:

• Group 1: 6 hours sleep
• Group 2: 7 hours sleep
• Group 3: 8 hours sleep

Test scores collected:

Group 1 (6 hours) [65, 67, 66, 64, 68 ]

Group 2 (7 hours) [70, 72, 71, 69, 73 ]

Group 3 (8 hours) [75, 78, 74, 76, 77]

Hypotheses

Manual Calculation :

Right tail is shaded because ANOVA is always a right-tailed test.

When ANOVA Says Significant

ANOVA tells us:

“At least one group is different.”

But it does NOT tell:

Which groups are different?

For that, we use:

Post-hoc tests like Tukey HSD.

Assumptions of ANOVA

1. Independent observations
2. Normally distributed groups
3. Equal variances (homogeneity)

If equal variance assumption fails, use Welch’s ANOVA.

When ANOVA Fails

Do NOT use ANOVA if:

• Data is categorical
• Groups are dependent
• Distribution is extremely skewed

In such cases use:

• Kruskal–Wallis test (non-parametric alternative)

What is Two-Way ANOVA?

Two-Way ANOVA tests:

• Effect of Factor A
• Effect of Factor B
• Interaction effect (A × B)

It answers:

• Does sleep affect scores?
• Does study method affect scores?
• Does the combination of sleep + study method matter?

Factor A: Sleep Hours

• A1 = 6 Hours
• A2 = 8 Hours

Factor B: Study Method

• B1 = Self Study
• B2 = Group Study

We collect 2 observations per cell.

Grand Total (T) = 592 Total Observations (N) = 8 Grand Mean = 592 / 8 = 74

Degrees of Freedom

• dfA = a − 1 = 2 − 1 = 1
• dfB = b − 1 = 2 − 1 = 1
• dfAB = (a − 1)(b − 1) = 1
• dfE = N − ab = 8 − 4 = 4
• dfTotal = N − 1 = 7

Note:

Error variance is zero because the dataset is perfectly structured (no randomness). In real data, SSE will never be zero.

Interpretation

Sleep has significant effect. Study method has significant effect. Interaction effect exists (sleep and method together influence performance).

If lines are not parallel → Interaction exists.

Special Cases in Two-Way ANOVA

• No Replication (One Observation per Cell) Interaction effect cannot be tested; only main effects are analyzed.
• Unequal Sample Sizes (Unbalanced Design) Requires Type II or Type III sum of squares for correct interpretation.
• Zero or Very Small Error Variance Produces extremely large F-values; often indicates artificial or perfectly structured data.
• Significant Interaction Present Main effects should not be interpreted independently.
• Violation of Equal Variance Assumption Use robust alternatives or data transformation.

Chi-Square Test

Chi-Square (χ²) is used for categorical data.

There are two main types:

1. Goodness of Fit Test
2. Test of Independence

Chi-Square Goodness of Fit Test

Purpose

Checks whether observed frequencies match expected frequencies.

Used when:

• One categorical variable
• Comparing observed distribution with theoretical distribution

Expected Frequency

Total = 100 Categories = 3

E=100/3=33.33E = 100 / 3 = 33.33E=100/3=33.33

Chi-Square Test of Independence

Purpose

Tests whether two categorical variables are independent.

Used with contingency tables.

Expected Frequencies

Assumptions

• Data is categorical
• Observations independent
• Expected frequency ≥ 5 (preferably)

Conclusions :

• Z-test and T-test help compare means to determine whether differences are statistically significant; Z-test is used when population variance is known, while T-test is used when it is unknown (more common in practice).
• One-Way ANOVA extends mean comparison to three or more groups, testing whether at least one group mean differs without increasing Type I error from multiple t-tests.
• Two-Way ANOVA evaluates the effect of two independent factors simultaneously and identifies whether an interaction between them influences the outcome.
• Chi-Square Goodness of Fit tests whether observed categorical data matches a theoretical distribution, while Chi-Square Test of Independence examines relationships between two categorical variables.
• Across all tests, correct interpretation depends on assumptions (normality, independence, equal variance, sufficient expected counts), proper hypothesis setup, and careful analysis of p-values and test statistics.