Introduction to Hypothesis Testing and Testing a Population Mean

Definition and Purpose: Hypothesis testing is a procedure that uses sample evidence and probability models to make statements regarding a population and subsequently check the validity of those statements.
The Three-Step Scientific Method Link: The process of statistics mirrors the scientific method: * Step 1: Make a statement regarding the nature of the population. * Step 2: Collect data (evidence) to test that statement. * Step 3: Analyze the data to assess the plausibility of the initial statement.
The Foundation: Statistics is built upon foundational steps that culminate in hypothesis testing, which is described as the "roof on the house" of the discipline.

The Null Hypothesis ( $H_0$ ): * Pronounced as " $H$ \text{ naught}". * Represents the statement to be tested. * Symbolizes "no change," "no effect," or "no difference." * It is the status quo—everything is exactly what it was thought to be. * Requirement: The null hypothesis is always an equality statement ( $p = ext{value}$ or $\mu = ext{value}$ ).
The Alternative Hypothesis ( $H_1$ or $H_a$ ): * Denoted as $H_1$ or sometimes $H_a$ . * The statement for which we are trying to find evidence to support. * Data is used to either support or disprove the alternative. * Assumption: We assume the null hypothesis is true throughout the test until evidence suggests otherwise.

Two-Tailed Test: * Symbolized by "not equal to" ( $\neq$ ). * Used when the researcher believes a change occurred but does not know the direction (greater or less than). * Confidence Intervals: These are effectively two-tailed hypothesis tests.
Left-Tailed Test: * Symbolized by "less than" ( $<$ ). * The focus is on values appearing to the left on a distribution curve.
Right-Tailed Test: * Symbolized by "greater than" ( $>$ ). * The focus is on values appearing to the right on a distribution curve.
Consistency of the Null: Regardless of the tail type, the null hypothesis ( $H_0$ ) is always set up the same way as an equality statement.

Proportion ( $p$ ): Used when dealing with percentages or probabilities. * Example: Medco Pharmaceutical antibiotic side effects (headaches). * $H_0: p = 0.02$ * $H_1: p eq 0.02$
Mean ( $\mu$ ): Used when dealing with averages (e.g., speed, time, or money). * Example: Test if a mean speed is greater than $60 \text{ mph}$ . * $H_0: \mu = 60$ * $H_1: \mu > 60$
Standard Deviation ( $\sigma$ ): Hypothesis tests can also be conducted on the variability of a population, following the same logic as means and proportions.

Theoretical Framework: In reality, the truth is unknown, but a conclusion must be made based on evidence.
Type I Error ( $\alpha$ ): * Occurs when you reject the null hypothesis when it is actually true. * Cheating Example: Accusing a friend of cheating when they were actually playing fair. This results in a "wrongfully accused" situation and is often more detrimental (likely leading to a mad or lost friend). * Hypothesis tests are typically built around controlling the probability of a Type I error.
Type II Error ( $\beta$ ): * Occurs when you fail to reject the null hypothesis (do not accuse) when the alternative is true (they were cheating). * Cheating Example: Your friend was cheating, but you failed to catch or accuse them. The friend is not mad; they "got away with it."
Correct Decisions: * Rejecting the null when the alternative is true. * Not rejecting the null when the null is true.

Level of Significance ( $\alpha$ ): * The probability of making a Type I error. * The same $\alpha$ used in confidence intervals. * Common Values: $0.05$ is the gold standard. However, it can be adjusted based on risk: * Low risk (e.g., Tylenol extra strength test): Might increase to $0.10$ . * High risk (e.g., Chemotherapy drug with fatal side effects): Might decrease to $0.01$ .
P-Value: * The probability that you "live in the tail" (the probability of obtaining the test statistic or one more extreme, assuming the null is true). * Decision Rule: If $P < \alpha$ , reject the null hypothesis.

The "No Accept" Rule: Statistically, we never accept the null hypothesis. We either "Reject the null" or "Fail to reject the null."
Phrasing for Rejection ( $P < \alpha$ ): "There is sufficient evidence to support the claim that [Alternative Hypothesis]."
Phrasing for Failure to Reject ( $P > \alpha$ ): "There is not sufficient evidence to support the claim that [Alternative Hypothesis]."
Connection to the Alternative: Conclusions are always phrased in terms of whether there is support for the alternative, not the truth of the null.

Before conducting a test, specific conditions must be met:
Randomness: The sample must be obtained via Simple Random Sample (SRS) or a randomized experimental design (random assignment to groups to reduce bias).
Normality: One of the following must be true: * The sample size is large ( $n \geq 30$ ). * The population is stated to be normally distributed. * The data shows a linear trend on a Normal Probability Plot (plotting $x$ values against $z$ -scores).
Independence: The sample size should be less than $5\%$ of the population ( $n < 0.05N$ ) so that sampling without replacement has no significant effect.

Step 0: Check conditions (Randomness, Normality, Independence).
Step 1: Determine the null ( $H_0$ ) and alternative ( $H_1$ ) hypotheses.
Step 2: Select the level of significance ( $\alpha$ ).
Step 3: Compute the test statistic. For a mean, we use the $t$ -statistic: * $t_0 = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}$ * Note: We use $t$ because we are estimating two parameters from the sample: the sample mean ( $\bar{x}$ ) and the sample standard deviation ( $s$ ).
Step 4: Determine the p-value (using Excel functions like T.DIST).
Step 5: Make a decision (Reject or Do Not Reject $H_0$ ).
Step 6: State the conclusion in context.

Scenario: The national mean student debt is $\$242.64$ . A sample of $n = 70$ individuals in Wyoming shows a mean of $\$176.46$ with a standard deviation of $\$163.56$ . We want to know if Wyoming's debt is less than the national average.
Step 0 (Conditions): * Random: Stated in the problem (Check). * Normality: $n = 70$ , which is $> 30$ (Check).
Step 1 (Hypotheses): * $H_0: \mu = 242.64$ * $H_1: \mu < 242.64$
Step 2 ( $\alpha$ ): $0.05$
Step 3 (Test Statistic): * $t_0 = \frac{176.46 - 242.64}{\frac{163.56}{\sqrt{70}}}$ * Numerator (negative) indicates the sample mean is less than the population mean. * The result is a negative $t$ -value, which is consistent with a left-tailed test.