(a) State the null hypothesis Ho and the alternative hypothesis H₁. H:0 H₁:0 (b) Determine the type of test statistic to use. (Choose one) (c) Find the value of the test statistic. (Round to three or more decimal places.) E H |x E 0=0 #0
(a) State the null hypothesis Ho and the alternative hypothesis H₁. H:0 H₁:0 (b) Determine the type of test statistic to use. (Choose one) (c) Find the value of the test statistic. (Round to three or more decimal places.) E H |x E 0=0 #0
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
![**Hypothesis Testing for Proportions**
A marriage counselor has traditionally observed that the proportion \( p \) of all married couples, for whom her communication program can prevent divorce, is 80%. After implementing recent changes, she claims that her program can prevent divorce in more than 80% of couples. In a random sample of 205 married couples who completed her program, 178 of them stayed together. We are tasked with determining if there is sufficient evidence to support the counselor's claim at the 0.10 level of significance.
**Steps for Hypothesis Testing:**
1. **State the Hypotheses:**
- **Null Hypothesis (\( H_0 \))**: \( p = 0.80 \)
- **Alternative Hypothesis (\( H_1 \))**: \( p > 0.80 \)
2. **Determine the Type of Test Statistic:**
For this scenario, we use a **z-test** to analyze population proportions.
3. **Calculate the Test Statistic:**
\[
z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}
\]
Where \( \hat{p} \) is the sample proportion, \( p_0 \) is the hypothesized population proportion (0.80), and \( n \) is the sample size (205).
4. **Find the p-value:**
The p-value is determined based on the calculated z-score and tells us the probability of observing a sample proportion as extreme as the one observed if the null hypothesis is true.
5. **Conclusion on Evidence:**
If the p-value is less than 0.10, there is sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis, supporting the counselor's claim that the program helps more than 80% of married couples.
**Note:** It's essential to carry out all intermediate calculations to at least three decimal places for accuracy. Use statistical software or z-tables to find the z-value and p-value.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9e284909-3fcc-4d39-81fd-66141de9ce6b%2Fb1bffacc-9956-46d3-97d9-b0bbe5649623%2Fiunewa5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Hypothesis Testing for Proportions**
A marriage counselor has traditionally observed that the proportion \( p \) of all married couples, for whom her communication program can prevent divorce, is 80%. After implementing recent changes, she claims that her program can prevent divorce in more than 80% of couples. In a random sample of 205 married couples who completed her program, 178 of them stayed together. We are tasked with determining if there is sufficient evidence to support the counselor's claim at the 0.10 level of significance.
**Steps for Hypothesis Testing:**
1. **State the Hypotheses:**
- **Null Hypothesis (\( H_0 \))**: \( p = 0.80 \)
- **Alternative Hypothesis (\( H_1 \))**: \( p > 0.80 \)
2. **Determine the Type of Test Statistic:**
For this scenario, we use a **z-test** to analyze population proportions.
3. **Calculate the Test Statistic:**
\[
z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}
\]
Where \( \hat{p} \) is the sample proportion, \( p_0 \) is the hypothesized population proportion (0.80), and \( n \) is the sample size (205).
4. **Find the p-value:**
The p-value is determined based on the calculated z-score and tells us the probability of observing a sample proportion as extreme as the one observed if the null hypothesis is true.
5. **Conclusion on Evidence:**
If the p-value is less than 0.10, there is sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis, supporting the counselor's claim that the program helps more than 80% of married couples.
**Note:** It's essential to carry out all intermediate calculations to at least three decimal places for accuracy. Use statistical software or z-tables to find the z-value and p-value.
Expert Solution
Step 1
It is given that
Population proportion, p = 80% = 0.80
Favourable cases, X = 178
Sample size, n = 205
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