Hypothesis Testing for a Population Mean (σ Unknown): Procedures, Examples, and Interpretation

Testing Claims About a Population Mean (σ Not Known)

Introduction to Hypothesis Testing for a Mean

Hypothesis testing for a population mean is a fundamental statistical procedure used to determine whether there is enough evidence in a sample to support a specific claim about the population mean, especially when the population standard deviation (σ) is unknown. This process is essential in scientific research, quality control, and many applied fields.

• Objective: Use a formal hypothesis test to evaluate claims about a population mean (μ) when σ is not known.

• Notation:

  • n = sample size

  • s = sample standard deviation

  • \bar{x} = sample mean

  • μ = population mean (the value specified in the null hypothesis H₀)

Requirements for the t-Test

• The sample must be a simple random sample.

• Either the population is normally distributed, or the sample size is large (n > 30) due to the Central Limit Theorem.

Test Statistic for a Mean (σ Unknown)

When the population standard deviation is unknown, the test statistic is calculated using the Student t-distribution:

$t = \frac{\bar{x} - \mu}{s / \sqrt{n}}$

• Degrees of freedom (df) = n - 1

• Critical values are obtained from the t-distribution table based on the chosen significance level (α) and df.

Step-by-Step Procedure for Hypothesis Testing

1. Establish Hypotheses

• Null Hypothesis (H₀): States that the population mean is equal to a specified value (e.g., μ = μ₀).

• Alternative Hypothesis (H₁): States the claim to be tested (e.g., μ ≠ μ₀, μ < μ₀, or μ > μ₀).

2. Choose the Significance Level (α)

• Common choices are 0.05 (5%) or 0.01 (1%).

• α represents the probability of making a Type I error (rejecting H₀ when it is true).

3. Compute the Test Statistic

• Calculate the sample mean (\bar{x}), sample standard deviation (s), and sample size (n).

• Use the formula for t (see above).

4. Find the P-Value

• The p-value is the probability of obtaining a test statistic as extreme as, or more extreme than, the observed value under H₀.

• Compare the p-value to α to make a decision.

5. Make a Decision

• If p-value < α, reject H₀.

• If p-value ≥ α, fail to reject H₀.

6. State the Conclusion in Context

• Restate the result in terms of the original claim, not just in terms of H₀ or H₁.

Worked Examples

Example 1: Three-Minute Hourglass Timer

Jo-jo tests whether shaking a sand timer affects the elapsed time. She collects 36 measurements and tests the claim that the average time is not 180 seconds.

• H₀: μ = 180 sec

• H₁ (Claim): μ ≠ 180 sec

• α: 0.05

• Calculate t using the sample data (not shown in detail here).

• Find the p-value and compare to α.

• Decision: Reject or fail to reject H₀.

• Conclusion: There (IS / IS NOT) enough evidence to (SUPPORT / REJECT) the claim that the time elapsed isn’t 180 seconds.

Example 2: Weight of Circulated Quarters

• Claim: The mean weight of circulated quarters is less than the production weight of 5.67 g.

• Sample Data: n = 50, \bar{x} = 5.6218 g, s = 0.0682 g, SE = 0.0096 g

• H₀: μ = 5.67 g

• H₁ (Claim): μ < 5.67 g

• α: 0.01

• Calculate t and p-value.

• Decision: Reject or fail to reject H₀.

• Conclusion: There (IS / IS NOT) enough evidence to (SUPPORT / REJECT) the claim that quarters in circulation weigh less than 5.67 g.

Example 3: Mean Age of Pennies in Circulation

• Claim: The mean age of a penny in circulation is 20 years.

• Sample Data: n = 800, \bar{x} = 21.153, s = 12.44

• H₀: μ = 20

• H₁ (Claim): μ ≠ 20

• α: 0.05

• Calculate t and p-value.

• Decision: Reject or fail to reject H₀.

• Conclusion: There (IS / IS NOT) enough evidence to (SUPPORT / REJECT) the claim that the average age is 20 years.

Confidence Intervals and Hypothesis Testing

A 95% confidence interval (CI) for the mean provides a range of plausible values for the population mean. If the value specified in H₀ is outside the CI, H₀ would be rejected at the 5% significance level.

• For the penny age example, the 95% CI is approximately (20.289, 22.015).

• Since 20 is within this interval, we would not reject H₀ at α = 0.05.

Summary Table: Steps in Hypothesis Testing for a Mean (σ Unknown)

Step	Description
1. Hypotheses	State H₀ and H₁ based on the claim
2. Significance Level	Choose α (e.g., 0.05 or 0.01)
3. Test Statistic	Calculate t using sample data
4. P-Value	Find probability of observed t under H₀
5. Decision	Compare p-value to α; reject or fail to reject H₀
6. Conclusion	State result in terms of the original claim

Key Terms and Concepts

• Null Hypothesis (H₀): The default assumption about a population parameter.

• Alternative Hypothesis (H₁): The claim being tested.

• Significance Level (α): Probability of Type I error.

• Test Statistic (t): Measures how far the sample mean is from the hypothesized mean in standard error units.

• P-Value: Probability of observing a result as extreme as the sample, assuming H₀ is true.

• Confidence Interval: Range of values likely to contain the population mean.