For each of the following scenarios: • Write down the rejection region, and • Compute the p-value for the statistic Using the p-value, conclude whether or not you reject Ho at level a = 0.05 1. where 0, o are unknown. and 2. where is unknown. iid X₁, X2,..., Xn N(μ, σ²) Ho: 0 10.0 vs. = Ha : 0 10.0 Ô = X s n 11.0 1.0 100 X₁, X2,..., Xn Ber(0)
For each of the following scenarios: • Write down the rejection region, and • Compute the p-value for the statistic Using the p-value, conclude whether or not you reject Ho at level a = 0.05 1. where 0, o are unknown. and 2. where is unknown. iid X₁, X2,..., Xn N(μ, σ²) Ho: 0 10.0 vs. = Ha : 0 10.0 Ô = X s n 11.0 1.0 100 X₁, X2,..., Xn Ber(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
I want to know the answer and detail step by step
![For each of the following scenarios:
- Write down the rejection region, and
- Compute the \( p \)-value for the statistic
- Using the \( p \)-value, conclude whether or not you reject \( H_0 \) at level \( \alpha = 0.05 \)
---
### 1.
\[ X_1, X_2, \ldots, X_n \overset{\text{iid}}{\sim} N(\mu, \sigma^2) \]
where \( \theta, \sigma \) are unknown.
\[ H_0 : \theta = 10.0 \quad \text{vs.} \quad H_a : \theta \neq 10.0 \]
and
\[
\hat{\theta} = \frac{\bar{X} \quad s^2 \quad n}{11.0 \quad 1.0 \quad 100}
\]
---
### 2.
\[ X_1, X_2, \ldots, X_n \overset{\text{iid}}{\sim} Ber(\theta) \]
where \( \theta \) is unknown.
\[ H_0 : \theta = 0.25 \quad \text{vs.} \quad H_a : \theta > 0.25 \]
and
\[
\hat{\theta} = \frac{\bar{X}}{n} \quad \frac{0.3}{49}
\]
For this one, use the CLT approximation for the sampling distribution of \( \hat{\theta} \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd7c2748e-5ba1-4f2f-bc7d-c6e1ab8bba12%2Fc48bac84-96b9-4bd4-8bbc-4168646015c4%2F66krizl_processed.png&w=3840&q=75)
Transcribed Image Text:For each of the following scenarios:
- Write down the rejection region, and
- Compute the \( p \)-value for the statistic
- Using the \( p \)-value, conclude whether or not you reject \( H_0 \) at level \( \alpha = 0.05 \)
---
### 1.
\[ X_1, X_2, \ldots, X_n \overset{\text{iid}}{\sim} N(\mu, \sigma^2) \]
where \( \theta, \sigma \) are unknown.
\[ H_0 : \theta = 10.0 \quad \text{vs.} \quad H_a : \theta \neq 10.0 \]
and
\[
\hat{\theta} = \frac{\bar{X} \quad s^2 \quad n}{11.0 \quad 1.0 \quad 100}
\]
---
### 2.
\[ X_1, X_2, \ldots, X_n \overset{\text{iid}}{\sim} Ber(\theta) \]
where \( \theta \) is unknown.
\[ H_0 : \theta = 0.25 \quad \text{vs.} \quad H_a : \theta > 0.25 \]
and
\[
\hat{\theta} = \frac{\bar{X}}{n} \quad \frac{0.3}{49}
\]
For this one, use the CLT approximation for the sampling distribution of \( \hat{\theta} \)
![### Example 3:
Consider random variables \(X_1, X_2, \ldots, X_n\) that are independent and identically distributed \(iid\) from a normal distribution \(N(\mu, \sigma^2)\), where \(\mu\) and \(\sigma^2\) are unknown. Let \(\theta = \sigma^2\), meaning we're focusing on testing the variance parameter.
- **Null Hypothesis (\(H_0\))**: \(\theta = 1.0\)
- **Alternative Hypothesis (\(H_a\))**: \(\theta \neq 1.0\)
The estimator for \(\theta\) is given by:
\[
\hat{\theta} = \frac{s^2}{n}
\]
With the sample variance \(s^2 = 1.2\) and sample size \(n = 100\).
---
### Example 4:
Consider random variables \(X_1, X_2, \ldots, X_n\) that are independent and identically distributed \(iid\) from an exponential distribution with parameter \(\theta\), denoted \(Exp(\theta)\), where \(\theta\) is unknown.
- **Null Hypothesis (\(H_0\))**: \(\theta = 2.0\)
- **Alternative Hypothesis (\(H_a\))**: \(\theta \neq 2.0\)
The estimator for \(\theta\) is given by:
\[
\hat{\theta} = \frac{\bar{X}}{n}
\]
With the sample mean \(\bar{X} = 2.2\) and sample size \(n = 100\).
---
These examples demonstrate hypothesis testing for variance and mean within normal and exponential distributions, respectively.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd7c2748e-5ba1-4f2f-bc7d-c6e1ab8bba12%2Fc48bac84-96b9-4bd4-8bbc-4168646015c4%2F8bkk4m_processed.png&w=3840&q=75)
Transcribed Image Text:### Example 3:
Consider random variables \(X_1, X_2, \ldots, X_n\) that are independent and identically distributed \(iid\) from a normal distribution \(N(\mu, \sigma^2)\), where \(\mu\) and \(\sigma^2\) are unknown. Let \(\theta = \sigma^2\), meaning we're focusing on testing the variance parameter.
- **Null Hypothesis (\(H_0\))**: \(\theta = 1.0\)
- **Alternative Hypothesis (\(H_a\))**: \(\theta \neq 1.0\)
The estimator for \(\theta\) is given by:
\[
\hat{\theta} = \frac{s^2}{n}
\]
With the sample variance \(s^2 = 1.2\) and sample size \(n = 100\).
---
### Example 4:
Consider random variables \(X_1, X_2, \ldots, X_n\) that are independent and identically distributed \(iid\) from an exponential distribution with parameter \(\theta\), denoted \(Exp(\theta)\), where \(\theta\) is unknown.
- **Null Hypothesis (\(H_0\))**: \(\theta = 2.0\)
- **Alternative Hypothesis (\(H_a\))**: \(\theta \neq 2.0\)
The estimator for \(\theta\) is given by:
\[
\hat{\theta} = \frac{\bar{X}}{n}
\]
With the sample mean \(\bar{X} = 2.2\) and sample size \(n = 100\).
---
These examples demonstrate hypothesis testing for variance and mean within normal and exponential distributions, respectively.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 4 steps
Recommended textbooks for you
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
Elementary Statistics: Picturing the World (7th E…
Statistics
ISBN:
9780134683416
Author:
Ron Larson, Betsy Farber
Publisher:
PEARSON
The Basic Practice of Statistics
Statistics
ISBN:
9781319042578
Author:
David S. Moore, William I. Notz, Michael A. Fligner
Publisher:
W. H. Freeman
Introduction to the Practice of Statistics
Statistics
ISBN:
9781319013387
Author:
David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:
W. H. Freeman