Formulating null and alternative hypotheses for one-sample tests.

📚 Quick Study Guide

• 🤔 Null Hypothesis (H₀): This is the statement being tested. It usually claims there's no effect or no difference. Think of it as the 'status quo'.
• 🧪 Alternative Hypothesis (H₁ or Ha): This is what you're trying to find evidence for. It contradicts the null hypothesis, suggesting there *is* an effect or difference.
• 🔢 Types of Alternative Hypotheses:
  • Two-tailed: The parameter is *not equal* to a specific value ($H_1: \mu \neq value$).
  • Right-tailed: The parameter is *greater than* a specific value ($H_1: \mu > value$).
  • Left-tailed: The parameter is *less than* a specific value ($H_1: \mu < value$).
• ✍️ Formulating Hypotheses: Always start with the research question, then express it mathematically as $H_0$ and $H_1$.
• 📊 One-Sample Tests: These tests compare a sample mean (or proportion) to a known population mean (or proportion). Examples include t-tests and z-tests.

Practice Quiz

1. A researcher believes the average height of adult males is greater than 5'10" (70 inches). Which of the following correctly states the null and alternative hypotheses?

   1. $H_0: \mu = 70, H_1: \mu < 70$
   2. $H_0: \mu = 70, H_1: \mu > 70$
   3. $H_0: \mu > 70, H_1: \mu = 70$
   4. $H_0: \mu \neq 70, H_1: \mu = 70$

2. A company claims its light bulbs last 1000 hours on average. You suspect the average lifespan is different. What are the null and alternative hypotheses?

   1. $H_0: \mu = 1000, H_1: \mu > 1000$
   2. $H_0: \mu = 1000, H_1: \mu < 1000$
   3. $H_0: \mu = 1000, H_1: \mu \neq 1000$
   4. $H_0: \mu \neq 1000, H_1: \mu = 1000$

3. A school principal wants to test if the proportion of students who eat school lunch is less than 60%. What are the correct hypotheses?

   1. $H_0: p = 0.6, H_1: p > 0.6$
   2. $H_0: p = 0.6, H_1: p < 0.6$
   3. $H_0: p < 0.6, H_1: p = 0.6$
   4. $H_0: p > 0.6, H_1: p = 0.6$

4. What type of alternative hypothesis is $H_1: \mu < 50$?

   1. Two-tailed
   2. Right-tailed
   3. Left-tailed
   4. Equal to

5. A researcher wants to prove that a new drug *decreases* blood pressure. Which alternative hypothesis is most appropriate?

   1. $H_1: \mu > value$
   2. $H_1: \mu = value$
   3. $H_1: \mu \neq value$
   4. $H_1: \mu < value$

6. If the null hypothesis is $H_0: \mu = 10$, which of the following is a valid two-tailed alternative hypothesis?

   1. $H_1: \mu > 10$
   2. $H_1: \mu < 10$
   3. $H_1: \mu \neq 10$
   4. $H_1: \mu = 10$

7. A study aims to determine if the average test score of students is significantly different from 75. The hypotheses are:

   1. $H_0: \mu = 75, H_1: \mu > 75$
   2. $H_0: \mu \neq 75, H_1: \mu = 75$
   3. $H_0: \mu = 75, H_1: \mu = 75$
   4. $H_0: \mu = 75, H_1: \mu \neq 75$