sample of size n = 16 is to be taken from a distribution that can reasonably be assumed to be Normal with a standard deviation σ of 100. The null hypothesis H0 : µ = 500 is to be tested against the alternative hypothesis H1 : µ = 520. What is the power of the .10 level test ? Choose the correct answer below and explain.
sample of size n = 16 is to be taken from a distribution that can reasonably be assumed to be Normal with a standard deviation σ of 100. The null hypothesis H0 : µ = 500 is to be tested against the alternative hypothesis H1 : µ = 520. What is the power of the .10 level test ? Choose the correct answer below and explain.
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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3. A sample of size n = 16 is to be taken from a distribution that can reasonably be assumed to be Normal with a standard deviation σ of 100. The null hypothesis H0 : µ = 500 is to be tested against the alternative hypothesis H1 : µ = 520.
What is the power of the .10 level test ? Choose the correct answer below and explain.
(A) .315 (B) .50 (C) .60 (D) .70
![### Sample Size Calculation
#### Question 10:
**Problem Statement:** Compute the sample size \( n \) needed for \( \mu_1 = 520 \), \( \beta = 0.2 \), and \( \alpha = 0.05 \).
\[ n = \sigma^2 \ast (z_\alpha + z_\beta)^2 / (\mu_1 - \mu_0)^2 \]
Given:
\[ \sigma = 100 \]
\[ z_\alpha = 1.645 \]
\[ z_\beta = 0.842 \]
\[ \mu_0 = 20^2 \]
Calculate \( n \):
\[ n = 100^2 \ast (1.645 + 0.842)^2 / (20^2) =? \]
Options:
(A) 16
(B) 32
(C) 68
(D) 155
#### Question 11:
**Problem Statement:** Compute the sample size \( n \) needed for \( \mu_1 = 510 \), \( \beta = 0.5 \), and \( \alpha = 0.05 \).
\[ n = \sigma^2 \ast (z_\alpha + z_\beta)^2 / (\mu_1 - \mu_0)^2 \]
Given:
\[ \sigma = 100 \]
\[ z_\alpha = 1.645 \]
\[ z_\beta = 0.000 \]
\[ \mu_0 = 20^2 \]
Calculate \( n \):
\[ n = 100^2 \ast (1.645 + 0.000)^2 / (20^2) =? \]
Options:
(A) 16
(B) 32
(C) 68
(D) 155
### Explanation of Equations and Constants:
- **Sample Size (\( n \))**: The number of observations in the sample required to achieve a certain level of statistical power.
- **Population Mean (\( \mu \))**: The average value of the population from which the sample is drawn.
- **Beta (\( \beta \))**: The probability of Type II error (failing to reject a false null hypothesis).
- **Alpha (\( \alpha \))**: The probability of Type I error (rejecting a true null hypothesis).
- **Standard Deviation (\( \sigma \))**: A](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8952a85a-85f1-4c32-8b45-c088a1def42e%2Fd7b92d8a-7f23-4a1e-b341-be57a55437b5%2Fp3phz7_processed.png&w=3840&q=75)
Transcribed Image Text:### Sample Size Calculation
#### Question 10:
**Problem Statement:** Compute the sample size \( n \) needed for \( \mu_1 = 520 \), \( \beta = 0.2 \), and \( \alpha = 0.05 \).
\[ n = \sigma^2 \ast (z_\alpha + z_\beta)^2 / (\mu_1 - \mu_0)^2 \]
Given:
\[ \sigma = 100 \]
\[ z_\alpha = 1.645 \]
\[ z_\beta = 0.842 \]
\[ \mu_0 = 20^2 \]
Calculate \( n \):
\[ n = 100^2 \ast (1.645 + 0.842)^2 / (20^2) =? \]
Options:
(A) 16
(B) 32
(C) 68
(D) 155
#### Question 11:
**Problem Statement:** Compute the sample size \( n \) needed for \( \mu_1 = 510 \), \( \beta = 0.5 \), and \( \alpha = 0.05 \).
\[ n = \sigma^2 \ast (z_\alpha + z_\beta)^2 / (\mu_1 - \mu_0)^2 \]
Given:
\[ \sigma = 100 \]
\[ z_\alpha = 1.645 \]
\[ z_\beta = 0.000 \]
\[ \mu_0 = 20^2 \]
Calculate \( n \):
\[ n = 100^2 \ast (1.645 + 0.000)^2 / (20^2) =? \]
Options:
(A) 16
(B) 32
(C) 68
(D) 155
### Explanation of Equations and Constants:
- **Sample Size (\( n \))**: The number of observations in the sample required to achieve a certain level of statistical power.
- **Population Mean (\( \mu \))**: The average value of the population from which the sample is drawn.
- **Beta (\( \beta \))**: The probability of Type II error (failing to reject a false null hypothesis).
- **Alpha (\( \alpha \))**: The probability of Type I error (rejecting a true null hypothesis).
- **Standard Deviation (\( \sigma \))**: A
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