A random sample of 92 observations produced a mean x = 25.8 and a standard deviation s = 2.4. a. Find a 95% confidence interval for µ. b. Find a 90% confidence interval for µ. c. Find a 99% confidence interval for u.
A random sample of 92 observations produced a mean x = 25.8 and a standard deviation s = 2.4. a. Find a 95% confidence interval for µ. b. Find a 90% confidence interval for µ. c. Find a 99% confidence interval for u.
MATLAB: An Introduction with Applications
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ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
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![### Confidence Interval Calculations
A random sample of 92 observations produced a mean (x̄) of 25.8 and a standard deviation (s) of 2.4.
#### Tasks:
a. Find a 95% confidence interval for μ.
b. Find a 90% confidence interval for μ.
c. Find a 99% confidence interval for μ.
### Step-by-Step Solution:
**1. Identify the parameters:**
- Sample mean (x̄) = 25.8
- Sample standard deviation (s) = 2.4
- Sample size (n) = 92
**2. Calculate the standard error of the mean (SEM):**
\[ \text{SEM} = \frac{s}{\sqrt{n}} = \frac{2.4}{\sqrt{92}} \]
**3. Use the appropriate z-score for each confidence level:**
- For a 95% confidence interval, the z-score is approximately 1.96.
- For a 90% confidence interval, the z-score is approximately 1.645.
- For a 99% confidence interval, the z-score is approximately 2.576.
**4. Calculate the margin of error (ME):**
\[ \text{ME} = z \times \text{SEM} \]
**5. Construct the confidence intervals:**
\[ \text{Confidence Interval} = x̄ \pm \text{ME} \]
**a. 95% Confidence Interval:**
- Calculate SEM:
\[ \text{SEM} = \frac{2.4}{\sqrt{92}} \approx 0.25 \]
- Calculate ME:
\[ \text{ME} = 1.96 \times 0.25 \approx 0.49 \]
- Confidence Interval:
\[ 25.8 \pm 0.49 = (25.31, 26.29) \]
**b. 90% Confidence Interval:**
- Calculate ME:
\[ \text{ME} = 1.645 \times 0.25 \approx 0.41 \]
- Confidence Interval:
\[ 25.8 \pm 0.41 = (25.39, 26.21) \]
**c. 99% Confidence Interval:**
- Calculate ME:
\[ \text{ME} = 2.576 \times 0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd210f98b-1208-4eea-a54e-043fbbd49724%2F081e0fd4-a91b-495e-a84b-c82ff3fb0806%2Fkca0i6f.png&w=3840&q=75)
Transcribed Image Text:### Confidence Interval Calculations
A random sample of 92 observations produced a mean (x̄) of 25.8 and a standard deviation (s) of 2.4.
#### Tasks:
a. Find a 95% confidence interval for μ.
b. Find a 90% confidence interval for μ.
c. Find a 99% confidence interval for μ.
### Step-by-Step Solution:
**1. Identify the parameters:**
- Sample mean (x̄) = 25.8
- Sample standard deviation (s) = 2.4
- Sample size (n) = 92
**2. Calculate the standard error of the mean (SEM):**
\[ \text{SEM} = \frac{s}{\sqrt{n}} = \frac{2.4}{\sqrt{92}} \]
**3. Use the appropriate z-score for each confidence level:**
- For a 95% confidence interval, the z-score is approximately 1.96.
- For a 90% confidence interval, the z-score is approximately 1.645.
- For a 99% confidence interval, the z-score is approximately 2.576.
**4. Calculate the margin of error (ME):**
\[ \text{ME} = z \times \text{SEM} \]
**5. Construct the confidence intervals:**
\[ \text{Confidence Interval} = x̄ \pm \text{ME} \]
**a. 95% Confidence Interval:**
- Calculate SEM:
\[ \text{SEM} = \frac{2.4}{\sqrt{92}} \approx 0.25 \]
- Calculate ME:
\[ \text{ME} = 1.96 \times 0.25 \approx 0.49 \]
- Confidence Interval:
\[ 25.8 \pm 0.49 = (25.31, 26.29) \]
**b. 90% Confidence Interval:**
- Calculate ME:
\[ \text{ME} = 1.645 \times 0.25 \approx 0.41 \]
- Confidence Interval:
\[ 25.8 \pm 0.41 = (25.39, 26.21) \]
**c. 99% Confidence Interval:**
- Calculate ME:
\[ \text{ME} = 2.576 \times 0
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