Module 8: One-Sided Tests, Proportions, and Error Types

One-Sided Hypothesis Testing

Concept Definition: A one-sided test focuses on a single direction in the alternative hypothesis ( $H_1$ ), either less than or greater than.
- Contrast with Two-Sided Tests: A two-sided test assesses if a parameter is equal to or not equal to a certain value (e.g., $\mu = 85$ vs. $\mu \ne 85$ ). A one-sided test specifically investigates if a parameter is less than a value (e.g., \mu < 80) or greater than a value (e.g., \mu > 85).
Graphical Representation (Shading Area):
- Two-Sided: The significance level, alpha ( $\alpha$ ), is divided equally between the two tails of the distribution.
- One-Sided: The entire alpha ( $\alpha$ ) is concentrated in only one tail (either left or right).
Determining the Correct Tail (Left vs. Right):
- Trick: Look at the direction of the alternative hypothesis ( $H_1$ ) symbol, treating it like an arrowhead.
 - If $H_1$ uses less than (<), the entire alpha goes to the left tail.
 - If $H_1$ uses greater than (>), the entire alpha goes to the right tail.
- (Note: The full theoretical background for this is extensive and not covered in depth in this course).
Example: Modified ACH Question 3 (Hypothetical Mineral Amount)
- Original Question: Mentioned

One-Sided Hypothesis Testing: Your "Directional" Detective

Imagine you're investigating a claim, but you're only interested if things have changed in a specific direction. That's the core idea behind a one-sided hypothesis test. It's like asking: "Is the new medicine better than the old one?" (not just "different") or "Is the machine producing less than it should?" (not just "not ideal").

The Core Question (Concept Definition)
- Your Goal: To determine if a population parameter (like a mean, $\mu$ ) is specifically less than or greater than a hypothesized value.
- How to Identify: Look for keywords in the problem that imply a direction: better than, worse than, increased, decreased, higher, smaller than, at least, at most.
- The Alternative Hypothesis ( $H_1$ ): This is where your directional question lives. It will always use either a less than (<) or greater than (>) symbol.
Choosing Your Lens: One-Sided vs. Two-Sided
- When to go One-Sided: Use this framework when your research question is purely directional. You only care if the evidence supports change in one specific way.
  - Think: "Is the mean less than 80?" or "Is the mean greater than 85?"
  - Example: Testing if a new production method reduces defects. You wouldn't care if it increases them, only if it decreases them.
- When to go Two-Sided: Use this framework when you just want to know if there's any difference at all.
  - Think: "Is the mean different from 85?" (meaning it could be greater or less).
  - Example: Testing if a new fertilizer changes crop yield. You'd be interested if it increases or decreases it, just that it's no longer the same.
Visualizing the "Risk Zone" (Graphical Representation)
- The Significance Level (Alpha, $\alpha$ ): This is your threshold for considering evidence significant. It's the maximum probability of making a Type I error (rejecting a true null hypothesis).
- Two-Sided: Your "risk zone" for significance is split into two equally small areas (tails) on both sides of the distribution. It's like having alarm bells at both ends.
  - Visualization: Imagine a normal curve; if $\alpha = 0.05$ , you'd shade $0.025$ on the far left and $0.025$ on the far right.
- One-Sided: Your entire $\alpha$ ("risk zone") is concentrated in only one tail. All your focus (and risk) is in the direction you're investigating.
  - Visualization: If $\alpha = 0.05$ and you're testing "less than," you'd shade the entire $0.05$ area on the far left tail. If "greater than," the entire $0.05$ on the far right tail. This means it's "easier" to find significance in one direction for the same $\alpha$ because you're concentrating all your "firepower" there.
Identifying the Correct Tail: Your Arrowhead Rule
- The "Trick" for Intuition: Look at the direction of the alternative hypothesis ( $H_1$ ) symbol (< or >).
 - If $H_1$ uses less than (<), it's pointing to the left. So, your entire $\alpha$ goes to the left tail. (Think: lower values are to the left on a number line).
 - If $H_1$ uses greater than (>), it's pointing to the right. So, your entire $\alpha$ goes to the right tail. (Think: higher values are to the right on a number line).
- Why this works: The alternative hypothesis defines the values for which you would reject the null hypothesis. If you believe the parameter is less than the hypothesized value, then very small sample means would lead you to reject. These small means are on the left side of the distribution.
Applying the Math (How to Process and Differentiate)
1. Formulate $H0$ and $H1$ : This is your critical first step to define your question. Make sure $H_1$ has the directional symbol.
2. Determine $\alpha$ : This is usually given or chosen (e.g., $0.05$ , $0.01$ ).
3. Calculate the Test Statistic: This number summarizes how far your sample data deviates from what $H_0$ predicts.
4. Find the Critical Value(s) OR p-value:
 - Critical Value Approach: For a one-sided test, you only find one critical value. If $H1$ is less than, you find the critical value that cuts off $\alpha$ in the left tail. If $H1$ is greater than, you find the critical value that cuts off $\alpha$ in the right tail.
 - P-value Approach: The p-value for a one-sided test is the area in one tail beyond your test statistic, in the direction of $H_1$ . You don't multiply it by 2.
5. Make a Decision:
 - Critical Value: Compare your test statistic to the single critical value. If your test statistic falls into the single shaded rejection region (the tail defined by $H1$ ), reject $H0$ .
 - P-value: If your one-sided p-value is less than or equal to $\alpha$ , reject $H_0$ .
Goal Function: When to Apply
- The goal of a one-sided test is to provide strong statistical evidence for a specific directional claim. If your objective is simply to detect any change, you use a two-sided test. If your objective is to prove an improvement or a detriment, a one-sided test is your tool.
Example: Modified ACH Question 3 (Hypothetical Mineral Amount)
- Currently, this refers to a placeholder. Let's imagine: "A mining company claims that a new drilling technique increases the average yield of a specific mineral beyond 500 lbs per ton. Historically, the yield was 500 lbs. Test at $\alpha = 0.05$ ."
- Your thought process:
  - Question: Does it increase yield? (Directional keyword: "increases")
  - $H_0$ : The average yield is $500$ lbs ( $\mu = 500$ ).
  - $H_1$ : The average yield is greater than $500$ lbs (\mu > 500).
  - Tail: Since $H_1$ uses greater than (>), it's a right-tailed test. The entire $\alpha = 0.05$ goes into the right tail.
  - Decision: You'd look for a sample