Stat Detective Worksheet
Stat Detective Case Files
Topic: Confidence Intervals
Formula: \(\bar{x} \pm z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}\)
Detective Name
Date
Analyze the following cases. Calculate the required confidence intervals using the population standard deviation provided. Round your final answers to one decimal place.
Case #101
The Commuter Crunch
A sample of 40 different commuters in a major city found that they spent an average of 32.5 minutes driving to work each day. If the population standard deviation for commute times is 6.2 minutes, find the 95% confidence interval for the average commute time \(\mu\) of all commuters in the city.
Calculations / Scratchpad
A
\(30.6 < \mu < 34.4\)
B
\(30.1 < \mu < 34.9\)
C
\(31.2 < \mu < 33.8\)
D
\(29.8 < \mu < 35.2\)
Case #102
The Bright Idea
A quality control inspector tested a sample of 50 long-life light bulbs and found they had an average lifespan of 1,240 hours. If the population standard deviation is known to be 85 hours, find the 99% confidence interval for the average lifespan \(\mu\) of all light bulbs of this type.
Calculations / Scratchpad
A
\(1,220.2 < \mu < 1,259.8\)
B
\(1,216.4 < \mu < 1,263.6\)
C
\(1,209.1 < \mu < 1,270.9\)
D
\(1,232.1 < \mu < 1,247.9\)
Stat Detective Answer Key
Answer Key
Stat Detective Case Files
The Z-Interval Formula
\[ \bar{x} \pm z_{\alpha/2} \cdot \left( \frac{\sigma}{\sqrt{n}} \right) \]
\(\bar{x}\) = Sample Mean
\(z_{\alpha/2}\) = Critical Value
\(\sigma\) = Population Std. Dev.
\(n\) = Sample Size
Case #101
The Commuter Crunch
Variables Identified
- \(n = 40\)
- \(\bar{x} = 32.5\)
- \(\sigma = 6.2\)
- Confidence Level = 95% \(\rightarrow z = 1.96\)
Correct Answer
Option A
\(30.6 < \mu < 34.4\)
Step-by-Step Solution
1. Calculate Standard Error: \(\frac{6.2}{\sqrt{40}} \approx 0.9803\)
2. Calculate Margin of Error (\(E\)): \(1.96 \cdot 0.9803 \approx 1.921\)
3. Lower Limit: \(32.5 - 1.9 \approx 30.6\)
4. Upper Limit: \(32.5 + 1.9 \approx 34.4\)
Case #102
The Bright Idea
Variables Identified
- \(n = 50\)
- \(\bar{x} = 1,240\)
- \(\sigma = 85\)
- Confidence Level = 99% \(\rightarrow z = 2.576\)
Correct Answer
Option C
\(1,209.1 < \mu < 1,270.9\)
Step-by-Step Solution
1. Calculate Standard Error: \(\frac{85}{\sqrt{50}} \approx 12.0208\)
2. Calculate Margin of Error (\(E\)): \(2.576 \cdot 12.0208 \approx 30.965\)
3. Lower Limit: \(1,240 - 30.9 = 1,209.1\)
4. Upper Limit: \(1,240 + 30.9 = 1,270.9\)
