To calculate the 80% confidence interval for the population mean \(\mu\), we will follow these steps:
- Calculate the Sample Mean (\(\bar{x}\)) and Sample Standard Deviation (s).
- Find the critical value (t) for 80% confidence level*.
- Calculate the Margin of Error.
- Determine the Confidence Interval.
Step 1: Calculate the Sample Mean and Sample Standard Deviation
Given the sample values:
- 87.1
- 69.8
- 94.3
- 94.7
- 76.3
- 88.4
- 95.3
- 87.8
- 109.5
- 85.8
- 102.5
- 85.6
Calculate the sample mean (\(\bar{x}\)):
\[ \bar{x} = \frac{\sum x_i}{n} = \frac{87.1 + 69.8 + 94.3 + 94.7 + 76.3 + 88.4 + 95.3 + 87.8 + 109.5 + 85.8 + 102.5 + 85.6}{12} = \frac{ 1,030.0 }{12} \approx 85.83 \]
Calculate the sample standard deviation (s):
\[ s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}} \]
Where:
- \(x_i\) are the sample values,
- \(\bar{x}\) is the sample mean,
- \(n\) is the number of samples.
Calculating the variance:
- First calculate \( (x_i - \bar{x})^2 \):
\[ \begin{align*} (87.1 - 85.83)^2 & \approx 1.6349 \ (69.8 - 85.83)^2 & \approx 258.0369 \ (94.3 - 85.83)^2 & \approx 70.0039 \ (94.7 - 85.83)^2 & \approx 78.6249 \ (76.3 - 85.83)^2 & \approx 90.7689 \ (88.4 - 85.83)^2 & \approx 6.5049 \ (95.3 - 85.83)^2 & \approx 87.4569 \ (87.8 - 85.83)^2 & \approx 3.8469 \ (109.5 - 85.83)^2 & \approx 566.5449 \ (85.8 - 85.83)^2 & \approx 0.0009 \ (102.5 - 85.83)^2 & \approx 284.7849 \ (85.6 - 85.83)^2 & \approx 0.0529 \ \end{align*} \]
The sum of these squared differences is: \[ 1.6349 + 258.0369 + 70.0039 + 78.6249 + 90.7689 + 6.5049 + 87.4569 + 3.8469 + 566.5449 + 0.0009 + 284.7849 + 0.0529 \approx 1,470.40 \]
Calculating the sample variance: \[ s^2 = \frac{1,470.40}{12 - 1} \approx 133.67 \quad \Rightarrow \quad s \approx 11.56 \]
Step 2: Find the Critical Value (t*) for 80% Confidence Level
For an 80% confidence level and \(n - 1 = 11\) degrees of freedom, we refer to the t-distribution table:
\[ t^* \approx 1.363 (approx) \]
Step 3: Calculate the Margin of Error
The margin of error \(E\) is calculated using the formula: \[ E = t^* \frac{s}{\sqrt{n}} = 1.363 \cdot \frac{11.56}{\sqrt{12}} \approx 1.363 \cdot 3.344 \approx 4.56 \]
Step 4: Determine the Confidence Interval
The 80% confidence interval is: \[ (\bar{x} - E, \bar{x} + E) = (85.83 - 4.56, 85.83 + 4.56) \approx (81.27, 90.39) \]
Therefore, the 80% confidence interval for the population mean \(\mu\) is: \[ \text{80% C.I.} = (81.27, 90.39) \]
So, the final answer is: \[ (81.27, 90.39) \]