Use the given information to find the number of degrees of freedom, the critical values x² and X, and the confidence interval estimate of σ. It is reasonable to assume that a simple random sample has been selected from a population with a normal distribution. Nicotine in menthol cigarettes 80% confidence; n = 20, s = 0.23 mg. Click the icon to view the table of Chi-Square critical values. df = 19 (Type a whole number.) x² = = 11.651 (Round to three decimal places as needed.) X²/2 = 27.204 (Round to three decimal places as needed.) The confidence interval estimate of o is 0.04 mg
Use the given information to find the number of degrees of freedom, the critical values x² and X, and the confidence interval estimate of σ. It is reasonable to assume that a simple random sample has been selected from a population with a normal distribution. Nicotine in menthol cigarettes 80% confidence; n = 20, s = 0.23 mg. Click the icon to view the table of Chi-Square critical values. df = 19 (Type a whole number.) x² = = 11.651 (Round to three decimal places as needed.) X²/2 = 27.204 (Round to three decimal places as needed.) The confidence interval estimate of o is 0.04 mg
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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7.20
![### Understanding Confidence Intervals with Chi-Square Distribution
When you have a simple random sample from a population with a normal distribution, you can calculate the confidence interval for the standard deviation, σ, using the Chi-Square distribution. This is especially useful in cases like determining nicotine levels in menthol cigarettes.
#### Problem Context
You are given:
- **80% Confidence Level** for nicotine in menthol cigarettes
- **Sample size (n)** = 20
- **Sample standard deviation (s)** = 0.23 mg
#### Objective
Find:
1. Number of degrees of freedom (df)
2. Critical Chi-Square values (χ²) for left (L) and right (R)
3. Confidence interval estimate for σ
#### Solution Steps
- **Degrees of Freedom (df):**
The degrees of freedom for a sample is calculated as n - 1.
\[ \text{df} = 20 - 1 = \boxed{19} \]
- **Critical Chi-Square Values:**
Using a Chi-Square distribution table corresponding to 80% confidence:
\[
\chi^2_L = \boxed{11.651} \quad (\text{rounded to three decimal places})
\]
\[
\chi^2_R = \boxed{27.204} \quad (\text{rounded to three decimal places})
\]
- **Confidence Interval for σ:**
The confidence interval for the true standard deviation is calculated using:
\[
\frac{(n-1) \times s^2}{\chi^2_R} < \sigma^2 < \frac{(n-1) \times s^2}{\chi^2_L}
\]
Solving gives the interval for σ:
\[
0.04 \, \text{mg} < \sigma < 0.09 \, \text{mg}
\]
By following these steps, it's possible to estimate the confidence interval for the standard deviation of nicotine content in menthol cigarettes. This method is widely used in statistics to infer population parameters when the underlying distribution is assumed to be normal.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6b463452-b960-4b00-bce0-3a69d9f467e2%2F84f4aafd-8ee4-495e-92ba-50fd444335c7%2F1hpxkh_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding Confidence Intervals with Chi-Square Distribution
When you have a simple random sample from a population with a normal distribution, you can calculate the confidence interval for the standard deviation, σ, using the Chi-Square distribution. This is especially useful in cases like determining nicotine levels in menthol cigarettes.
#### Problem Context
You are given:
- **80% Confidence Level** for nicotine in menthol cigarettes
- **Sample size (n)** = 20
- **Sample standard deviation (s)** = 0.23 mg
#### Objective
Find:
1. Number of degrees of freedom (df)
2. Critical Chi-Square values (χ²) for left (L) and right (R)
3. Confidence interval estimate for σ
#### Solution Steps
- **Degrees of Freedom (df):**
The degrees of freedom for a sample is calculated as n - 1.
\[ \text{df} = 20 - 1 = \boxed{19} \]
- **Critical Chi-Square Values:**
Using a Chi-Square distribution table corresponding to 80% confidence:
\[
\chi^2_L = \boxed{11.651} \quad (\text{rounded to three decimal places})
\]
\[
\chi^2_R = \boxed{27.204} \quad (\text{rounded to three decimal places})
\]
- **Confidence Interval for σ:**
The confidence interval for the true standard deviation is calculated using:
\[
\frac{(n-1) \times s^2}{\chi^2_R} < \sigma^2 < \frac{(n-1) \times s^2}{\chi^2_L}
\]
Solving gives the interval for σ:
\[
0.04 \, \text{mg} < \sigma < 0.09 \, \text{mg}
\]
By following these steps, it's possible to estimate the confidence interval for the standard deviation of nicotine content in menthol cigarettes. This method is widely used in statistics to infer population parameters when the underlying distribution is assumed to be normal.
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