Construct a 95% confidence interval for the population mean, μ. Assume the population has a normal distribution. A sample of 20 college students had mean annual earnings of $3120 with a standard deviation of $677. contents OA. ($1324,$1567) B. ($2657,$2891) C. ($2803,$3437) D. ($2135,$2567) e Resourc r Success or Success media Library F estly cloudy Ask my instructor H 4 Clear all Check answer
Construct a 95% confidence interval for the population mean, μ. Assume the population has a normal distribution. A sample of 20 college students had mean annual earnings of $3120 with a standard deviation of $677. contents OA. ($1324,$1567) B. ($2657,$2891) C. ($2803,$3437) D. ($2135,$2567) e Resourc r Success or Success media Library F estly cloudy Ask my instructor H 4 Clear all Check answer
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![### Constructing a 95% Confidence Interval for Population Mean
**Scenario:**
You are given a task to construct a 95% confidence interval for the population mean, denoted as \( \mu \). Assume the population follows a normal distribution. A sample of 20 college students yields the following statistics:
- Mean annual earnings: $3120
- Standard deviation: $677
**Options:**
You are provided with four potential confidence intervals. Select the correct one:
- A. ($1324, $1567)
- B. ($2657, $2991)
- C. ($2803, $3437)
- D. ($2135, $2567)
**Steps to Solve:**
1. **Identify Given Information:**
- Sample size (n): 20
- Sample mean (\( \bar{x} \)): $3120
- Sample standard deviation (s): $677
2. **Determine the Confidence Level:**
Since the confidence level is 95%, the corresponding z-value (critical value) for a two-tailed test is approximately 1.96 (from z-tables).
3. **Calculate the Margin of Error (ME):**
The formula for the margin of error in this context is given by:
\[ ME = z \times \left(\frac{s}{\sqrt{n}}\right) \]
4. **Substitute Given Values into Formula:**
\[
ME = 1.96 \times \left(\frac{677}{\sqrt{20}}\right)
\]
5. **Compute Margin of Error:**
\[
ME = 1.96 \times \left(\frac{677}{4.47}\right) \approx 1.96 \times 151.45 \approx 296.84
\]
6. **Determine the Confidence Interval:**
The confidence interval can be found using:
\[
\left( \bar{x} - ME, \bar{x} + ME \right)
\]
7. **Substitute Values:**
\[
\left( 3120 - 296.84, 3120 + 296.84 \right) \approx \left( 2823.16, 3416.84 \right)
\]
Therefore, the correct answer from the options given is:
- C. ($2803](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2a348127-0366-4d6a-8a8b-87b3575e425d%2F7306fb8f-20e0-44ba-a67d-3dda8d36c56f%2Fu5c4e6l_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Constructing a 95% Confidence Interval for Population Mean
**Scenario:**
You are given a task to construct a 95% confidence interval for the population mean, denoted as \( \mu \). Assume the population follows a normal distribution. A sample of 20 college students yields the following statistics:
- Mean annual earnings: $3120
- Standard deviation: $677
**Options:**
You are provided with four potential confidence intervals. Select the correct one:
- A. ($1324, $1567)
- B. ($2657, $2991)
- C. ($2803, $3437)
- D. ($2135, $2567)
**Steps to Solve:**
1. **Identify Given Information:**
- Sample size (n): 20
- Sample mean (\( \bar{x} \)): $3120
- Sample standard deviation (s): $677
2. **Determine the Confidence Level:**
Since the confidence level is 95%, the corresponding z-value (critical value) for a two-tailed test is approximately 1.96 (from z-tables).
3. **Calculate the Margin of Error (ME):**
The formula for the margin of error in this context is given by:
\[ ME = z \times \left(\frac{s}{\sqrt{n}}\right) \]
4. **Substitute Given Values into Formula:**
\[
ME = 1.96 \times \left(\frac{677}{\sqrt{20}}\right)
\]
5. **Compute Margin of Error:**
\[
ME = 1.96 \times \left(\frac{677}{4.47}\right) \approx 1.96 \times 151.45 \approx 296.84
\]
6. **Determine the Confidence Interval:**
The confidence interval can be found using:
\[
\left( \bar{x} - ME, \bar{x} + ME \right)
\]
7. **Substitute Values:**
\[
\left( 3120 - 296.84, 3120 + 296.84 \right) \approx \left( 2823.16, 3416.84 \right)
\]
Therefore, the correct answer from the options given is:
- C. ($2803
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