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MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![### Hypothesis Testing: Energy Levels and Diet
**Question 3 SLO**
**Objective:** Identify the null hypothesis, alternative hypothesis, test statistic, p-value, initial and final conclusion.
**Problem Statement:** Test the claim that less than 45% of the people following a particular diet will experience increased energy. Of 81 randomly selected subjects who followed the diet, 42 noticed an increase in their energy level.
---
To solve this, we need to follow the steps of hypothesis testing. Here's the breakdown:
1. **Formulate Hypotheses:**
- **Null Hypothesis (H₀):** \( p \geq 0.45 \)
- This indicates that 45% or more of the people following the diet will experience increased energy.
- **Alternative Hypothesis (H₁):** \( p < 0.45 \)
- This indicates that less than 45% of the people following the diet will experience increased energy.
2. **Collect Data:**
- Sample size (n): 81
- Number of subjects experiencing increased energy (x): 42
- Sample proportion (\( \hat{p} \)): \( \frac{42}{81} \approx 0.5185 \)
3. **Calculate the Test Statistic:**
- The test statistic for a proportion is calculated using the formula:
\[
z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}
\]
where \( p_0 \) is the hypothesized population proportion (0.45).
4. **Determine the p-value:**
- The p-value indicates the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true. This value helps us decide if we should reject the null hypothesis.
5. **Make a Decision:**
- Compare the p-value with the chosen significance level (α), often set at 0.05.
- If p-value < α, reject the null hypothesis.
- Otherwise, do not reject the null hypothesis.
6. **Conclusion:**
- State your initial conclusion based on the p-value.
- Provide final conclusion: whether there is sufficient evidence to support the claim that less than 45% of the people following the diet will experience increased energy.
---
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Transcribed Image Text:### Hypothesis Testing: Energy Levels and Diet
**Question 3 SLO**
**Objective:** Identify the null hypothesis, alternative hypothesis, test statistic, p-value, initial and final conclusion.
**Problem Statement:** Test the claim that less than 45% of the people following a particular diet will experience increased energy. Of 81 randomly selected subjects who followed the diet, 42 noticed an increase in their energy level.
---
To solve this, we need to follow the steps of hypothesis testing. Here's the breakdown:
1. **Formulate Hypotheses:**
- **Null Hypothesis (H₀):** \( p \geq 0.45 \)
- This indicates that 45% or more of the people following the diet will experience increased energy.
- **Alternative Hypothesis (H₁):** \( p < 0.45 \)
- This indicates that less than 45% of the people following the diet will experience increased energy.
2. **Collect Data:**
- Sample size (n): 81
- Number of subjects experiencing increased energy (x): 42
- Sample proportion (\( \hat{p} \)): \( \frac{42}{81} \approx 0.5185 \)
3. **Calculate the Test Statistic:**
- The test statistic for a proportion is calculated using the formula:
\[
z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}
\]
where \( p_0 \) is the hypothesized population proportion (0.45).
4. **Determine the p-value:**
- The p-value indicates the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true. This value helps us decide if we should reject the null hypothesis.
5. **Make a Decision:**
- Compare the p-value with the chosen significance level (α), often set at 0.05.
- If p-value < α, reject the null hypothesis.
- Otherwise, do not reject the null hypothesis.
6. **Conclusion:**
- State your initial conclusion based on the p-value.
- Provide final conclusion: whether there is sufficient evidence to support the claim that less than 45% of the people following the diet will experience increased energy.
---
By
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