Hypothesis Testing for the Mean (Sigma Unknown) – t-Test Applications

Hypothesis Testing with One Sample

Introduction to Hypothesis Testing

Hypothesis testing is a fundamental statistical method used to make inferences about population parameters based on sample data. In this chapter, we focus on testing hypotheses about the population mean when the population standard deviation is unknown, utilizing the t-distribution.

Hypothesis Testing for the Mean (Sigma Unknown)

Finding Critical Values in a t-Distribution

When the population standard deviation (σ) is unknown, the t-distribution is used to determine critical values for hypothesis tests involving the mean. The process involves:

• Specifying the level of significance (α): Common choices are 0.05, 0.01, or 0.10.

• Identifying the degrees of freedom (df): Calculated as df = n - 1, where n is the sample size.

• Finding the critical value(s): Use the t-table (Table 5 in Appendix B) to locate the appropriate value based on the test type:

  • Left-tailed test: Use the "One Tail" column with a negative sign.

  • Right-tailed test: Use the "One Tail" column with a positive sign.

  • Two-tailed test: Use the "Two Tails" column for both negative and positive critical values.

Example: Finding Critical Values

• Left-tailed test: For α = 0.05 and n = 21, df = 20. The critical value is negative and found in the "One Tail" column for df = 20.

• Right-tailed test: For α = 0.01 and n = 17, df = 16. The critical value is positive and found in the "One Tail" column for df = 16.

• Two-tailed test: For α = 0.05 and n = 26, df = 25. The critical values are both positive and negative, found in the "Two Tails" column for df = 25.

t-Test for a Mean (Sigma Unknown)

The t-test for a mean is used when the population standard deviation is unknown. The test statistic is calculated as follows:

$t = \frac{\overline{x} - \mu_0}{s / \sqrt{n}}$

• \( \overline{x} \): Sample mean

• \( \mu_0 \): Hypothesized population mean

• \( s \): Sample standard deviation

• \( n \): Sample size

The t-test is appropriate when:

• The sample is random.

• The population is normally distributed, or the sample size is sufficiently large (Central Limit Theorem).

Using the t-Test for Mean (Sigma Unknown)

The steps for conducting a t-test are:

1. State the null and alternative hypotheses.

2. Choose the significance level (α).

3. Calculate the test statistic using the formula above.

4. Find the critical value(s) from the t-distribution table.

5. Compare the test statistic to the critical value(s) or use the P-value approach.

6. Make a decision: reject or fail to reject the null hypothesis.

Example: Hypothesis Testing Using a Rejection Region

Scenario: A used car dealer claims the mean transaction price is at least $29,100. A sample of 14 cars has a mean of $27,177 and a standard deviation of $3,250. Test at α = 0.05.

• Step 1: Hypotheses: H0: μ ≥ 29,100, Ha: μ < 29,100

• Step 2: Left-tailed test, df = 13

• Step 3: Find critical value from t-table for α = 0.05, df = 13

• Step 4: Calculate t-statistic

• Step 5: Compare t to critical value; if t is in the rejection region, reject H0

Conclusion: There is enough evidence at the 5% level to reject the dealer’s claim.

Example: Hypothesis Testing Using Rejection Regions

Scenario: An industrial company claims the mean pH of river water is 6.8. A sample of 39 has a mean of 6.7 and standard deviation of 0.35. Test at α = 0.05.

• Step 1: Hypotheses: H0: μ = 6.8, Ha: μ ≠ 6.8

• Step 2: Two-tailed test, df = 38

• Step 3: Find critical values from t-table for α = 0.05, df = 38

• Step 4: Calculate t-statistic

• Step 5: Compare t to critical values; if t is not in the rejection region, fail to reject H0

Conclusion: There is not enough evidence at the 5% level to reject the company’s claim.

Using P-Values with t-Tests

The P-value approach compares the probability of observing a test statistic as extreme as, or more extreme than, the observed value under the null hypothesis. The steps are:

1. Calculate the t-statistic.

2. Find the P-value using statistical software or tables.

3. Compare the P-value to α:

   • If P-value ≤ α, reject H0.

   • If P-value > α, fail to reject H0.

Example: Using P-Values with t-Tests

Scenario: DMV claims mean wait time is less than 14 minutes. Sample of 10 people: mean = 13, s = 3.5. Test at α = 0.10.

• Step 1: Hypotheses: H0: μ = 14, Ha: μ < 14

• Step 2: Calculate t-statistic

• Step 3: Find P-value

• Step 4: Compare P-value to α; if P-value > α, fail to reject H0

Conclusion: There is not enough evidence at the 10% level to support the office’s claim.

Summary Table: t-Test Decision Process

Step	Description
1	State hypotheses (H0, Ha)
2	Choose significance level (α)
3	Calculate test statistic (t)
4	Find critical value(s) or P-value
5	Make decision: reject or fail to reject H0

Additional info: The t-test is robust to moderate departures from normality, especially as sample size increases. Technology (such as statistical calculators or software) can be used to compute P-values and critical values efficiently.