Confidence Interval Problem with Unknown Sigma

Overview

• Focus on confidence interval problems where Sigma is unknown.
• This involves raw data processing to find necessary statistics.

Problem Setup

• Sample Data: GPA for 12 randomly selected college students.
• Tasks:
  • Find the sample mean.
  • Find the sample standard deviation.

Steps for Solution

Calculating Sample Mean

1. Software Setup
   • Use Excel or LibreOffice.
   • Enter formula: =AVERAGE(...) to calculate the sample mean.
2. Result
   • Sample Mean (( \bar{X} )): 2.3

Calculating Sample Standard Deviation

1. Using Software
   • Enter formula: =STDEV(...) to calculate the sample standard deviation.
2. Result
   • Sample Standard Deviation (s): 1.1901 (rounded to four decimal places)

Constructing Confidence Interval

• Formula: ( \bar{X} \pm t \times S_{\bar{X}} )
• Use T-distribution since Sigma is unknown._

Calculate ( S_{\bar{X}} )_

• Formula: ( S_{\bar{X}} = \frac{s}{\sqrt{n}} )
• Calculation:
  • ( n = 12 )
  • ( S_{\bar{X}} = \frac{1.1901}{\sqrt{12}} = 0.3436 ) (rounded)

Finding T-Value

1. Determine Degrees of Freedom
   • ( df = n - 1 = 11 )
2. Confidence Level
   • 95% confidence interval → 0.025 in each tail.
3. T-Score Calculation
   • Use software: statistical analysis tools.
   • T-value: 2.201 (rounded to three decimal places)

Final Confidence Interval Calculation

• Computation:
  • ( 2.201 \times 0.3436 = 0.756 ) (rounded)
  • Confidence Interval: ( 2.3 \pm 0.756 )
  • Interval: [1.544, 3.056] (rounded to three decimal places)

Conclusion

• This method enables the construction of a confidence interval when Sigma is unknown, utilizing sample data, and T-distribution principles.