Section 10.3

Hypothesis Tests for a Population Mean

Use Student's t-distribution when σ is unknown to test claims about population means.

1

Model Requirements

Random sample or randomized experiment
Population is normally distributed OR
(sample ≤ 5% of population)
2

t-Test Statistic

Degrees of Freedom:

3

Robustness of the t-Test

Robust: The t-test works well even with minor departures from normality, provided there are no outliers.

Sample Size Guidelines

n < 15:Use t-test only if data is normal with no outliers
15 ≤ n < 30:Use t-test if no outliers and only mild skewness
n ≥ 30:Use t-test even with skewness (CLT applies), but watch for outliers
4

Classical & P-Value Approaches

Classical Approach

Compare to critical values from t-table with . Reject if in rejection region.

P-Value Approach

If P-value < , reject . Use t-distribution with .

5

t-Test Calculator

Hypothesis Test Calculator for Population Mean (t-test)

Perform t-tests when σ is unknown

Hypotheses

Null:
Alternative:
Degrees of Freedom:
df = 34

Calculations

Sample Mean
Standard Error
Critical Value
Test Statistic

t-Distribution Visualization (df = 34)

-1.6921.6920t₀ = -3.82Reject H₀Reject H₀Fail to Reject H₀

Classical Approach

Critical Region: t < -1.692 or t > 1.692
t₀ = -3.8168 is in rejection region

P-Value Approach

P-value = 0.0007
P-value < α = 0.05

Reject H₀

At the α = 0.05 significance level, there is sufficient evidence to conclude that the population mean differs from 98.6.

!

Common Pitfalls

Using Z instead of t

When σ is unknown, always use the t-distribution, not Z.

Ignoring Outliers

Outliers can severely affect the sample mean and standard deviation, invalidating the test.

Wrong Degrees of Freedom

For a one-sample t-test, df = n - 1, not n.

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