Hypothesis Testing: Procedures for Proportions, Means, and Standard Deviations

Hypothesis Testing

Introduction to Hypothesis Testing

Hypothesis testing is a fundamental statistical procedure used to make inferences about population parameters based on sample data. It involves formulating two competing statements about a population and using probability to determine which statement is better supported by the data.

• Hypothesis: A proposition or statement about a population parameter (such as mean, proportion, or standard deviation).

• Hypothesis Testing: A process that uses sample evidence and probability to test statements about population characteristics.

Proper rounding is important in hypothesis testing to ensure accuracy in calculations and conclusions.

Key Terms and Concepts

• Null Hypothesis (H0): The default or initial claim about a population parameter, usually representing no effect or no difference.

• Alternative Hypothesis (H1 or Ha): The competing claim, representing a new effect, difference, or relationship.

• Level of Significance (\( \alpha \)): The probability of making a Type I error (rejecting a true null hypothesis). Common values are 0.05, 0.01, or 0.10.

• Type I Error: Rejecting the null hypothesis when it is actually true. Probability is \( \alpha \).

• Type II Error: Failing to reject the null hypothesis when the alternative hypothesis is true. Probability is \( \beta \).

• Test Statistic: A standardized value calculated from sample data, used to decide whether to reject H0.

• Critical Value: The threshold value that the test statistic is compared to, based on \( \alpha \) and the sampling distribution.

• P-value: The probability, assuming H0 is true, of obtaining a result as extreme or more extreme than the observed result.

Types of Hypothesis Tests

• Two-tailed Test: Tests for deviations in both directions (e.g., H1: \( \mu \neq \mu_0 \)).

• Right-tailed Test: Tests for an increase (e.g., H1: \( \mu > \mu_0 \)).

• Left-tailed Test: Tests for a decrease (e.g., H1: \( \mu < \mu_0 \)).

General Steps in Hypothesis Testing

1. State the hypotheses: Formulate H0 and H1.

2. Select the significance level (\( \alpha \)): Choose the probability threshold for Type I error.

3. Determine the appropriate test statistic and its distribution: (z, t, or chi-square, depending on the parameter and sample size).

4. Calculate the test statistic from the sample data.

5. Find the critical value(s) or p-value: Based on \( \alpha \) and the test type.

6. Make a decision: Reject or fail to reject H0 based on the comparison.

7. State the conclusion in context.

Hypothesis Tests for a Population Proportion

Testing a Population Proportion

When testing claims about a population proportion, the sampling distribution of the sample proportion is approximately normal if the sample size is large enough.

• Test Statistic (z):

$z_0 = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$

• \( \hat{p} \): Sample proportion

• \( p_0 \): Claimed population proportion under H0

• \( n \): Sample size

Example: Testing a Drug Manufacturer's Claim

• Claim: More than 94% of patients are healed within 8 weeks (\( p > 0.94 \)).

• Sample: 213 out of 224 patients healed (\( \hat{p} = 213/224 \approx 0.9509 \)).

• Hypotheses: \( H_0: p = 0.94 \), \( H_1: p > 0.94 \) (right-tailed test).

• Significance Level: \( \alpha = 0.01 \).

• Critical Value: z = 2.33 (area to the right is 0.01).

• Test Statistic:

$z_0 = \frac{0.9509 - 0.94}{\sqrt{\frac{0.94 \times 0.06}{224}}} \approx 0.69$

• Since 0.69 < 2.33, do not reject H0.

• Conclusion: There is not enough evidence at the 1% significance level to support the manufacturer's claim.

P-value Approach

• Calculate the p-value corresponding to the test statistic.

• If p-value < \( \alpha \), reject H0; otherwise, do not reject H0.

• In the example, p-value = 0.2451 > 0.01, so do not reject H0.

Connection to Binomial Distribution

• For small samples, the binomial distribution can be used to test hypotheses about proportions.

• For large samples, the normal approximation is appropriate.

Hypothesis Tests for a Population Mean

Testing a Population Mean

When testing claims about a population mean, the t-distribution is used if the population standard deviation is unknown and the sample size is small (n < 30). For large samples, the normal distribution may be used.

• Test Statistic (t):

$t_0 = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}$

• \( \bar{x} \): Sample mean

• \( \mu_0 \): Claimed population mean under H0

• \( s \): Sample standard deviation

• \( n \): Sample size

Example: SAT Critical Reading Scores

• Claim: Students who learn English and another language simultaneously score lower than the general population (\( \mu < 501 \)).

• Sample: n = 100, \( \bar{x} = 485 \), s = 116.

• Hypotheses: \( H_0: \mu = 501 \), \( H_1: \mu < 501 \) (left-tailed test).

• Significance Level: \( \alpha = 0.10 \).

• Critical Value: t = -1.290.

• Test Statistic:

$t_0 = \frac{485 - 501}{\frac{116}{\sqrt{100}}} = \frac{-16}{11.6} \approx -1.38$

• Since -1.38 < -1.290, reject H0.

• Conclusion: There is sufficient evidence at the 10% significance level to support the claim that these students score lower.

P-value Approach

• Calculate the p-value for the test statistic.

• If p-value < \( \alpha \), reject H0.

• In the example, p-value = 0.0855 < 0.10, so reject H0.

Statistical vs. Practical Significance

• Statistical Significance: The result is unlikely to have occurred by chance, according to the chosen \( \alpha \).

• Practical Significance: The result is large enough to be meaningful in real-world terms.

Hypothesis Tests for a Population Standard Deviation

Testing a Population Standard Deviation

To test claims about a population standard deviation (or variance), the chi-square (\( \chi^2 \)) distribution is used. The steps are similar to those for means and proportions, but the test statistic and critical values come from the chi-square distribution.

• Test Statistic (chi-square):

$\chi^2_0 = \frac{(n-1)s^2}{\sigma_0^2}$

• \( s^2 \): Sample variance

• \( \sigma_0^2 \): Claimed population variance under H0

• \( n \): Sample size

The critical values depend on the degrees of freedom (df = n - 1) and the significance level \( \alpha \).

Summary of Hypothesis Testing Procedures

Parameter	Test Statistic	Distribution	When to Use
Proportion (p)	$z_0 = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$	Standard Normal (z)	Large n, categorical data
Mean (\( \mu \))	$t_0 = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}$	t-distribution	Unknown \( \sigma \), small n
Standard Deviation (\( \sigma \))	$\chi^2_0 = \frac{(n-1)s^2}{\sigma_0^2}$	Chi-square (\( \chi^2 \))	Variance/SD claims

Additional Notes

• Both the traditional (critical value) and p-value approaches are important and may be required in assignments and exams.

• Confidence intervals can be used to support conclusions from hypothesis tests.

• Tables for critical values (z, t, chi-square) are found in the textbook appendix or provided with test questions.

Additional info: The notes reference the use of Minitab for p-value calculations and confidence intervals, and emphasize the importance of understanding both statistical and practical significance.