Exam 3 Review: Hypothesis Testing, Confidence Intervals, and Sampling Distributions

Hypothesis Testing

Types of Errors and Symbols

In hypothesis testing, we use specific symbols and terminology to describe errors and test statistics. Understanding these is crucial for interpreting results and making decisions.

• Type I Error (α): Rejecting the null hypothesis when it is actually true. The probability of a Type I error is denoted by $\alpha$.

• Type II Error (β): Failing to reject the null hypothesis when it is actually false. The probability of a Type II error is denoted by $\beta$.

• Test Statistic (z, t): Used to determine whether to reject the null hypothesis. Common test statistics include the z-score and t-score.

• Sample Mean ($\bar{x}$): The average value in a sample, used in test statistics.

• Sample Standard Deviation (s): Measures the spread of sample data.

• Population Standard Deviation ($\sigma$): Measures the spread of population data.

• Sample Proportion ($\hat{p}$): The proportion of successes in a sample.

Example: If $\alpha = 0.05$, there is a 5% chance of committing a Type I error.

Decision Rules in Hypothesis Testing

Decisions in hypothesis testing are based on the comparison of the test statistic to critical values or p-values.

• Reject $H_0$: If the test statistic falls in the rejection region or if the p-value is less than $\alpha$.

• Fail to Reject $H_0$: If the test statistic does not fall in the rejection region or if the p-value is greater than $\alpha$.

Example: For a two-tailed test with $\alpha = 0.05$, reject $H_0$ if $|z| > 1.96$.

Sampling Distributions and Estimation

Sampling Distributions

A sampling distribution is the probability distribution of a given statistic based on a random sample. It is fundamental for making inferences about populations.

• Sample Mean Distribution: If the population is normal or the sample size is large ($n \geq 30$), the sampling distribution of the sample mean is approximately normal.

• Standard Error: The standard deviation of the sampling distribution. For the mean, $SE = \frac{\sigma}{\sqrt{n}}$.

Example: For $\sigma = 10$ and $n = 25$, $SE = \frac{10}{5} = 2$.

Confidence Intervals

Confidence intervals provide a range of values within which the population parameter is likely to fall, with a certain level of confidence.

• Formula for Mean (Known $\sigma$): $\bar{x} \pm z^* \frac{\sigma}{\sqrt{n}}$

• Formula for Mean (Unknown $\sigma$): $\bar{x} \pm t^* \frac{s}{\sqrt{n}}$

• Formula for Proportion: $\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

Example: For $\bar{x} = 50$, $s = 8$, $n = 16$, and $t^* = 2.131$ (from t-table for 15 df, 95% CI): $50 \pm 2.131 \times \frac{8}{4} = 50 \pm 4.262$

Testing Hypotheses for Two Variables

Comparing Two Means

When comparing two means, we use different formulas depending on whether samples are independent or dependent.

• Independent Samples: $t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$

• Dependent Samples (Paired): $t = \frac{\bar{d}}{\frac{s_d}{\sqrt{n}}}$, where $\bar{d}$ is the mean of the differences.

Example: For paired data, calculate the difference for each pair, then compute $\bar{d}$ and $s_d$.

Comparing Two Proportions

To test the difference between two proportions:

• Test Statistic: $z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}}$ where $\hat{p}$ is the pooled proportion.

Example: If $\hat{p}_1 = 0.6$, $n_1 = 100$, $\hat{p}_2 = 0.5$, $n_2 = 120$, calculate pooled $\hat{p}$ and substitute.

Critical Values and Statistical Tables

z-Table, t-Table, and Chi-Square Table

Statistical tables are used to find critical values for hypothesis tests and confidence intervals.

Distribution	Purpose	Key Parameters
z-Table	Critical values for normal distribution	Mean, standard deviation
t-Table	Critical values for t-distribution (small samples)	Degrees of freedom
Chi-Square Table	Critical values for chi-square tests	Degrees of freedom

Example: For a two-tailed test at $\alpha = 0.05$, the critical z-value is $\pm 1.96$.

Assumptions in Hypothesis Testing

Key Assumptions

Correct application of statistical tests requires certain assumptions:

• Random sampling

• Independence of observations

• Normality of population (for t-tests)

• Equal variances (for some two-sample tests)

Example: For a two-sample t-test, check that both samples are independent and variances are equal.

Formulating Hypotheses

Null and Alternative Hypotheses

Every hypothesis test begins with formulating null ($H_0$) and alternative ($H_a$) hypotheses.

• Null Hypothesis ($H_0$): States that there is no effect or difference.

• Alternative Hypothesis ($H_a$): States that there is an effect or difference.

• One-tailed vs. Two-tailed: One-tailed tests look for a difference in a specific direction; two-tailed tests look for any difference.

Example: $H_0: \mu_1 = \mu_2$ vs. $H_a: \mu_1 \neq \mu_2$ (two-tailed)

Application: Interpreting Results and Making Decisions

Using p-values and Critical Regions

Decisions are made by comparing the p-value to the significance level $\alpha$ or by checking if the test statistic falls in the critical region.

• If $p < \alpha$, reject $H_0$.

• If $p \geq \alpha$, fail to reject $H_0$.

Example: If $p = 0.03$ and $\alpha = 0.05$, reject $H_0$.

Summary Table: Key Symbols and Their Meanings

Symbol	Meaning
$\alpha$	Type I error rate (significance level)
$\beta$	Type II error rate
$\bar{x}$	Sample mean
$s$	Sample standard deviation
$\sigma$	Population standard deviation
$\hat{p}$	Sample proportion
$z$	z-score (standard normal test statistic)
$t$	t-score (Student's t test statistic)

Additional info:

• Some content inferred from context and standard statistics curriculum, such as formulas and definitions.

• Tables reconstructed to summarize critical values and symbols.