Hypothesis Testing for One Sample: Proportion and Mean

Hypothesis Testing for One Sample

The Language of Hypothesis Testing

Hypothesis testing is a fundamental procedure in inferential statistics, used to make decisions about population parameters based on sample data. The process involves formulating two competing statements about a population parameter and using sample evidence to determine which statement is more plausible.

• Null Hypothesis (H0): A statement of no change, effect, or difference. It is assumed true until evidence suggests otherwise.

• Alternative Hypothesis (H1): The statement for which we seek evidence. It represents a change, effect, or difference.

Types of tests:

• Two-tailed test: H0: parameter = value; H1: parameter ≠ value

• Left-tailed test: H0: parameter = value; H1: parameter < value

• Right-tailed test: H0: parameter = value; H1: parameter > value

Example: If a researcher wants to know if the mean lifetime of a light bulb is less than 500 hours, the hypotheses are:

• H0: μ = 500

• H1: μ < 500

Type I and Type II Errors

When conducting hypothesis tests, two types of errors can occur:

• Type I Error (α): Rejecting H0 when it is actually true.

• Type II Error (β): Failing to reject H0 when H1 is actually true.

The level of significance (α) is the probability of making a Type I error. Lowering α reduces the chance of a Type I error but increases the chance of a Type II error.

Stating Conclusions

Conclusions are stated in terms of the alternative hypothesis. We never "accept" the null hypothesis; we either reject it or fail to reject it.

• "There is sufficient evidence to conclude that..." (if H0 is rejected)

• "There is not sufficient evidence to conclude that..." (if H0 is not rejected)

Hypothesis Tests for a Population Proportion

Logic of Hypothesis Testing

Hypothesis testing evaluates whether observed sample results are unlikely under the assumption that the null hypothesis is true. If so, the result is statistically significant, and H0 is rejected.

• Classical Approach: Compare the test statistic to a critical value.

• P-value Approach: The P-value is the probability of observing a result as extreme as the sample, assuming H0 is true. If P-value < α, reject H0.

• Confidence Interval Approach: If the hypothesized value is not within the confidence interval, reject H0.

Testing Hypotheses about a Population Proportion

To test hypotheses about a population proportion, follow these steps:

1. Verify Assumptions:

   • Simple random sample or randomized experiment

   • np0 ≥ 10 and nq0 ≥ 10

   • Sampled values are independent (n ≤ 0.05N)

2. State Hypotheses: Structure as two-tailed, left-tailed, or right-tailed.

3. Select Level of Significance (α): Common values are 0.05, 0.01, or 0.10.

4. Calculate Test Statistic:

   • Test statistic for proportion: $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 q_0}{n}}}$

5. Find P-value or Critical Value: Use statistical tables or calculator functions.

6. Make a Decision: Compare P-value to α or test statistic to critical value.

7. State Conclusion: Interpret in context of the problem.

Hypothesis Tests for a Population Mean

Testing Hypotheses about a Mean

When testing hypotheses about a population mean, the t-distribution is used if the population standard deviation is unknown and the sample size is small. The steps are similar to those for proportions, with the following test statistic:

• Test statistic for mean: $t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}$

Where:

• \( \bar{x} \) = sample mean

• \( \mu_0 \) = hypothesized population mean

• \( s \) = sample standard deviation

• \( n \) = sample size

Degrees of freedom (df) = n - 1.

Using the t-Distribution Table

The t-distribution table provides critical values for various significance levels and degrees of freedom. These values are used to determine rejection regions for hypothesis tests involving means.

Statistical vs. Practical Significance

Statistical significance means that the observed effect is unlikely to have occurred by chance, given the null hypothesis. Practical significance considers whether the effect size is large enough to be meaningful in real-world terms. A result can be statistically significant but not practically important if the difference is too small to matter in practice.

Summary Table: Types of Errors in Hypothesis Testing

Decision	H0 True	H1 True
Reject H0	Type I Error (α)	Correct Decision
Do Not Reject H0	Correct Decision	Type II Error (β)

Key Formulas

• Test Statistic for Proportion: $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 q_0}{n}}}$

• Test Statistic for Mean: $t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}$

• Confidence Interval for Proportion: $\hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

• Confidence Interval for Mean: $\bar{x} \pm t_{\alpha/2, df} \frac{s}{\sqrt{n}}$

Additional info: The t-distribution is used instead of the normal distribution when the population standard deviation is unknown and the sample size is small (n < 30). As the sample size increases, the t-distribution approaches the normal distribution.