Confidence Quest Guide
Confidence Quest
Name: __________________
Date: ___________________
Mastering the true mean using the t-distribution.
The Formula Anatomy
When we don't know the population standard deviation (\(\sigma\)) and our sample size is small (\(n < 30\)), we use the t-distribution to build our interval:
\[ \bar{x} \pm t_{\alpha/2} \cdot \left( \frac{s}{\sqrt{n}} \right) \]
\(\bar{x}\)
Sample Mean: The "best guess" for the middle of our interval.
\(t_{\alpha/2}\)
Critical Value: Based on the confidence level and degrees of freedom (\(df = n-1\)).
\(s\)
Sample Std. Deviation: Measures the spread within our sample.
\(n\)
Sample Size: The total number of observations collected.
1
Soda Filling Station (95% Confidence)
"A soft drink bottler samples 16 bottles off the line. The mean volume is 20.2 oz. with a sample standard deviation of 0.60 oz. Find the 95% confidence interval of the true mean volume."
Step 1: Identify the Variables
\(n = 16\)
\(\bar{x} = 20.2\)
\(s = 0.60\)
\(C = 95\%\)
Step 2: Find Degrees of Freedom & Critical Value
Calculate Degrees of Freedom: \(df = n - 1 = 16 - 1 = \mathbf{15}\)
Using a t-table for \(df = 15\) at a 95% confidence level (\(\alpha = 0.05\)):
\(t_{\alpha/2} = 2.131\)
Step 3: Calculate the Margin of Error (\(E\))
\[ E = t_{\alpha/2} \cdot \left( \frac{s}{\sqrt{n}} \right) \] \[ E = 2.131 \cdot \left( \frac{0.60}{\sqrt{16}} \right) \] \[ E = 2.131 \cdot \left( \frac{0.60}{4} \right) = 2.131 \cdot 0.15 \] \[ E \approx 0.32 \]
Step 4: Construct & Interpret
Lower Limit: \(20.2 - 0.32 = \mathbf{19.88}\)
Upper Limit: \(20.2 + 0.32 = \mathbf{20.52}\)
Conclusion:
We are 95% confident that the true mean volume of all soda bottles is between 19.88 oz. and 20.52 oz.
2
Battery Lifespan (99% Confidence)
"A manufacturer tests 9 premium batteries. The sample mean life is 450 minutes with a sample standard deviation of 30 minutes. Find the 99% confidence interval of the true mean life."
Step 1: Identify the Variables
\(n = 9\)
\(\bar{x} = 450\)
\(s = 30\)
\(C = 99\%\)
Step 2: Find Degrees of Freedom & Critical Value
Calculate Degrees of Freedom: \(df = n - 1 = 9 - 1 = \mathbf{8}\)
Using a t-table for \(df = 8\) at a 99% confidence level (\(\alpha = 0.01\)):
\(t_{\alpha/2} = 3.355\)
Step 3: Calculate the Margin of Error (\(E\))
\[ E = 3.355 \cdot \left( \frac{30}{\sqrt{9}} \right) \] \[ E = 3.355 \cdot \left( \frac{30}{3} \right) = 3.355 \cdot 10 \] \[ E = 33.55 \]
Step 4: Construct & Interpret
Lower Limit: \(450 - 33.55 = \mathbf{416.45}\)
Upper Limit: \(450 + 33.55 = \mathbf{483.55}\)
Conclusion:
We are 99% confident that the true mean battery life is between 416.45 minutes and 483.55 minutes.
Statistical Note
Why does the 99% interval feel "wider" than the 95% one? Higher confidence requires a larger margin of error to ensure the true mean is captured. As confidence levels increase, so does the critical value (\(t^*\)), pushing the limits further out!
