f) Test the claim that µ = 15. Provide a sketch of the Critical Value. Evaluate the Test Statistic and provide the p- value.
f) Test the claim that µ = 15. Provide a sketch of the Critical Value. Evaluate the Test Statistic and provide the p- value.
MATLAB: An Introduction with Applications
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ISBN:9781119256830
Author:Amos Gilat
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Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Topic Video
Question
![**Data Set Analysis**
Use the following data set: **40, 33, 77, 12, 23, 56, 23, 19, 29** (in minutes). Assume the data is approximately bell-shaped and is a sample.
In this dataset, each number represents a time duration recorded in minutes. The assumption that the data is "approximately bell-shaped" suggests that it follows a normal distribution, which is typically symmetrical, with most data points clustering around the mean. This characteristic is useful for statistical analysis, helping to apply techniques like z-scores and standard deviation. Understanding the shape of the data can assist in deriving insights about the average performance and variability within this sample.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F31273fa4-0ef0-4537-a09e-f2fe1ffff523%2F6d74cd89-9973-4e45-94f6-8dea95370867%2Fi1ykeqi_processed.png&w=3840&q=75)
Transcribed Image Text:**Data Set Analysis**
Use the following data set: **40, 33, 77, 12, 23, 56, 23, 19, 29** (in minutes). Assume the data is approximately bell-shaped and is a sample.
In this dataset, each number represents a time duration recorded in minutes. The assumption that the data is "approximately bell-shaped" suggests that it follows a normal distribution, which is typically symmetrical, with most data points clustering around the mean. This characteristic is useful for statistical analysis, helping to apply techniques like z-scores and standard deviation. Understanding the shape of the data can assist in deriving insights about the average performance and variability within this sample.
![**Hypothesis Testing for Mean (\( \mu \))**
**Objective:**
- Test the claim that the population mean (\( \mu \)) is equal to 15.
- Sketch the distribution indicating the Critical Value.
- Evaluate the Test Statistic.
- Provide the p-value.
**Steps:**
1. **State the Hypotheses:**
- Null Hypothesis (\( H_0 \)): \( \mu = 15 \)
- Alternative Hypothesis (\( H_a \)): \( \mu \neq 15 \)
2. **Select the Significance Level (\( \alpha \)):**
- Common choices for \( \alpha \) are 0.05, 0.01, etc.
3. **Sketch the Distribution:**
- Draw a normal distribution curve.
- Mark the Critical Value(s) on the curve.
- The Critical Value delimiters will depend on your significance level.
4. **Calculate the Test Statistic:**
- Use the formula for the test statistic (e.g., Z or T) based on the sample data.
- Plug in your sample mean, population mean, standard deviation, and sample size into the formula.
5. **Determine the p-value:**
- Locate the Test Statistic on the standard normal distribution.
- The p-value is the probability that the test statistic is as extreme as, or more extreme than, the observed value under the null hypothesis.
6. **Make a Decision:**
- If the p-value < \( \alpha \), reject the null hypothesis (\( H_0 \)).
- If the p-value ≥ \( \alpha \), do not reject the null hypothesis (\( H_0 \)).
**Conclusion:**
- Based on the results, conclude whether there is sufficient evidence to reject the claim that \( \mu = 15 \).
**Visual Explanation:**
- A graphical representation can help visualize where the test statistic and critical values lie in relation to the significance level on the curve.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F31273fa4-0ef0-4537-a09e-f2fe1ffff523%2F6d74cd89-9973-4e45-94f6-8dea95370867%2F3csmtrl_processed.png&w=3840&q=75)
Transcribed Image Text:**Hypothesis Testing for Mean (\( \mu \))**
**Objective:**
- Test the claim that the population mean (\( \mu \)) is equal to 15.
- Sketch the distribution indicating the Critical Value.
- Evaluate the Test Statistic.
- Provide the p-value.
**Steps:**
1. **State the Hypotheses:**
- Null Hypothesis (\( H_0 \)): \( \mu = 15 \)
- Alternative Hypothesis (\( H_a \)): \( \mu \neq 15 \)
2. **Select the Significance Level (\( \alpha \)):**
- Common choices for \( \alpha \) are 0.05, 0.01, etc.
3. **Sketch the Distribution:**
- Draw a normal distribution curve.
- Mark the Critical Value(s) on the curve.
- The Critical Value delimiters will depend on your significance level.
4. **Calculate the Test Statistic:**
- Use the formula for the test statistic (e.g., Z or T) based on the sample data.
- Plug in your sample mean, population mean, standard deviation, and sample size into the formula.
5. **Determine the p-value:**
- Locate the Test Statistic on the standard normal distribution.
- The p-value is the probability that the test statistic is as extreme as, or more extreme than, the observed value under the null hypothesis.
6. **Make a Decision:**
- If the p-value < \( \alpha \), reject the null hypothesis (\( H_0 \)).
- If the p-value ≥ \( \alpha \), do not reject the null hypothesis (\( H_0 \)).
**Conclusion:**
- Based on the results, conclude whether there is sufficient evidence to reject the claim that \( \mu = 15 \).
**Visual Explanation:**
- A graphical representation can help visualize where the test statistic and critical values lie in relation to the significance level on the curve.
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