• Chemists often need to estimate population parameters from sample data
• Examples: average molecular weight, reaction yield, concentration of a solution
• Interval estimation provides a range of plausible values for these parameters
• Confidence Intervals (CI) are commonly used in chemical analysis

A confidence interval in chemistry:

• Is calculated from experimental data
• Has a specified level of confidence (e.g., 95%)
• Provides a range likely to contain the true value of a chemical property

\[\text{CI} = \text{Measured Value} \pm \text{Margin of Error}\]

1. Point estimate (e.g., mean concentration \(\bar{x}\))
2. Critical value (e.g., \(t\) value for small samples)
3. Standard error of the estimate

For a mean concentration:

\[\text{CI} = \bar{x} \pm t \cdot \frac{s}{\sqrt{n}}\]

Where: - \(\bar{x}\) is the sample mean concentration - \(t\) is the critical value from t-distribution - \(s\) is the sample standard deviation - \(n\) is the number of measurements

• A 95% CI means that if we repeat the experiment many times, about 95% of the intervals would contain the true concentration
• It does not mean there’s a 95% probability that the true concentration is in the interval
• Wider intervals indicate less precision in the measurement, narrower intervals indicate more precision

Suppose we want to estimate the concentration of a copper sulfate solution:

• Number of measurements: n = 5
• Sample mean: \(\bar{x} = 0.1025\) mol/L
• Sample standard deviation: s = 0.0015 mol/L
• Confidence level: 95% (t ≈ 2.776 for df = 4)

## 95% CI: 0.1006 to 0.1044 mol/L

calculate_chem_ci <- function(x_bar, s, n, conf_level = 0.95) {
  degrees_of_freedom <- n - 1
  t_score <- qt((1 + conf_level) / 2, df = degrees_of_freedom)
  margin_error <- t_score * (s / sqrt(n))
  
  ci_lower <- x_bar - margin_error
  ci_upper <- x_bar + margin_error
  
  return(c(ci_lower, ci_upper))
}

# Example: Copper sulfate solution concentration
x_bar <- 0.1025  # mol/L
s <- 0.0015      # mol/L
n <- 5

ci <- calculate_chem_ci(x_bar, s, n)
cat("95% CI:", round(ci[1], 4), "to", round(ci[2], 4), "mol/L")

## 95% CI: 0.1006 to 0.1044 mol/L

• Interval estimation is crucial for reporting uncertainty in chemical measurements
• CIs help in assessing the reliability of experimental results
• Understanding factors affecting CI width aids in designing better experiments
• Always interpret CIs in the context of the specific chemical analysis
• Consider practical significance along with statistical significance in chemistry