15) Do a hypothesis for the following, make sure to include and label all steps: Test the claim that for the adult population of one town, the mean salary is $ 27,000. If a sample of 25 has a mean of $ 23,500 and a sample standard deviation of $ 11,200. Use a significance level of 0.05. amazon 99+
15) Do a hypothesis for the following, make sure to include and label all steps: Test the claim that for the adult population of one town, the mean salary is $ 27,000. If a sample of 25 has a mean of $ 23,500 and a sample standard deviation of $ 11,200. Use a significance level of 0.05. amazon 99+
MATLAB: An Introduction with Applications
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Author:Amos Gilat
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Chapter1: Starting With Matlab
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![### Hypothesis Testing Example
#### Problem Statement:
Test the claim that for the adult population of one town, the mean salary is $27,000. If a sample of 25 has a mean of $23,500 and a sample standard deviation of $11,200. Use a significance level of 0.05.
#### Step-by-Step Solution:
1. **State the Hypotheses:**
- Null Hypothesis (H₀): μ = $27,000 (The mean salary for the population is $27,000)
- Alternative Hypothesis (H₁): μ ≠ $27,000 (The mean salary for the population is not $27,000)
2. **Set the Significance Level:**
- α = 0.05
3. **Calculate the Test Statistic:**
- We will use the t-test for the sample mean since the population standard deviation is unknown and the sample size is small (n < 30).
Formula:
\[
t = \frac{\bar{x} - \mu}{s / \sqrt{n}}
\]
Where:
- \(\bar{x}\) = sample mean = $23,500
- \(\mu\) = population mean = $27,000
- \(s\) = sample standard deviation = $11,200
- \(n\) = sample size = 25
\[
t = \frac{23,500 - 27,000}{11,200 / \sqrt{25}}
\]
\[
t = \frac{-3,500}{2,240}
\]
\[
t ≈ -1.56
\]
4. **Determine the Degrees of Freedom:**
- df = n - 1 = 25 - 1 = 24
5. **Find the Critical Value:**
- Using a t-table or calculator for a two-tailed test at α = 0.05 significance level and 24 degrees of freedom.
- Critical t-value ≈ ±2.064
6. **Make the Decision:**
- Compare the calculated t-value with the critical t-value.
- If \(-t_{critical} \leq t \leq t_{critical}\), fail to reject H₀.
- Here, \(-2.064 ≤](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff6ad1cd7-e883-4155-acb4-5513247aaa08%2Ff030993f-0980-4375-871b-1b7b276aea4f%2F8oc232u_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Hypothesis Testing Example
#### Problem Statement:
Test the claim that for the adult population of one town, the mean salary is $27,000. If a sample of 25 has a mean of $23,500 and a sample standard deviation of $11,200. Use a significance level of 0.05.
#### Step-by-Step Solution:
1. **State the Hypotheses:**
- Null Hypothesis (H₀): μ = $27,000 (The mean salary for the population is $27,000)
- Alternative Hypothesis (H₁): μ ≠ $27,000 (The mean salary for the population is not $27,000)
2. **Set the Significance Level:**
- α = 0.05
3. **Calculate the Test Statistic:**
- We will use the t-test for the sample mean since the population standard deviation is unknown and the sample size is small (n < 30).
Formula:
\[
t = \frac{\bar{x} - \mu}{s / \sqrt{n}}
\]
Where:
- \(\bar{x}\) = sample mean = $23,500
- \(\mu\) = population mean = $27,000
- \(s\) = sample standard deviation = $11,200
- \(n\) = sample size = 25
\[
t = \frac{23,500 - 27,000}{11,200 / \sqrt{25}}
\]
\[
t = \frac{-3,500}{2,240}
\]
\[
t ≈ -1.56
\]
4. **Determine the Degrees of Freedom:**
- df = n - 1 = 25 - 1 = 24
5. **Find the Critical Value:**
- Using a t-table or calculator for a two-tailed test at α = 0.05 significance level and 24 degrees of freedom.
- Critical t-value ≈ ±2.064
6. **Make the Decision:**
- Compare the calculated t-value with the critical t-value.
- If \(-t_{critical} \leq t \leq t_{critical}\), fail to reject H₀.
- Here, \(-2.064 ≤
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