Statistics - Confidence Intervals

Confidence Interval Overview

• Confidence intervals provide a range of plausible values for an unknown population parameter, on which we place a high degree of confidence that it includes the true value.
• They expand point estimates to interval estimates, offering a range for our estimates of a parameter, typically the mean (BC).

Confidence Interval for the Population Mean

Large Sample Size (n >= 30)

• Central Limit Theorem: For large sample sizes, the sampling distribution of the sample mean ($ar{x}$) is approximately normal.
• The formula for a 95% confidence interval for the mean is: ar{x} ext{ } ext{±} Z_{0.025} rac{s}{ ext{√}n}, where:
  • Z_{0.025} is the critical value from the z-distribution (1.96 for 95% confidence).
  • s is the sample standard deviation.

Example Calculation

• Sample of 80 shops:
  • Mean cost of repair ($ar{x}$) = $472.36
  • Standard deviation ($s$) = $62.35
• Confidence interval:
  472.36 ext{ ± } 1.96 rac{62.35}{ ext{√}80}
  ightarrow [458.7, 486.02]
• Interpretation: We are 95% confident that the true mean cost is within this interval.

Understanding Confidence

• Misinterpretation: It is incorrect to say there's a 95% probability that the true mean falls within the interval. The parameter (BC) is fixed, not random.
• Randomness applies to the sampling process that generates different intervals.

Calculating Confidence Levels

• For intervals derived from independent samples, about 95% will contain the true mean ($BC$).
• For confidence levels:
  • 90% confidence: Critical value = 1.645
  • 99% confidence: Critical value = 2.575

Precision in Confidence Intervals

• Factors affecting interval width:
  1. Sample Standard Deviation (s): Larger values widen the interval.
  2. Confidence Level: Higher confidence levels widen the interval (Z alpha increases).
  3. Sample Size (n): Larger sample sizes narrow the interval. Width is inversely proportional to $ ext{√}n$.

Sample Size Determination

• To achieve a specific maximum error ($W$) in a 95% confidence interval:
  n = rac{(Z_{0.025} s)}{W}^2

Smaller Sample Size (n < 30)

• If the sample size is small and the population distribution is normal:
• Use the t-distribution instead of the normal distribution:
  • The t-distribution is similar but has heavier tails, accommodating more variability.
• Confidence interval formula adjusts:
  ar{x} ext{ } ext{±} t_{n-1} rac{s}{ ext{√}n}
• Where t_{n-1} is the critical t-value with (n-1) degrees of freedom.

Example with t-Distribution

• Sample mean ($ar{x}$) = 61,492, sample standard deviation (s) = 3,035, sample size (n) = 10:
• Degrees of freedom (df) = 9.
• Critical value for 95% confidence ($t_{0.025}$, 9 df) = 2.262.
• Confidence interval calculation:
  61,492 ext{ } ext{±} 2.262 rac{3,035}{ ext{√}10}
  ightarrow [59,321.04, 63,662.96]

One-Sided Confidence Intervals

• For one-sided bounds, replace Z{0.025} with Z{ ext{α}} in:
  • Lower confidence bound: ar{x} - Z_{ ext{α}} rac{s}{ ext{√}n}
  • Upper confidence bound: ar{x} + Z_{ ext{α}} rac{s}{ ext{√}n}

Confidence Interval for Population Proportion

Large Samples

• Proportions can also be estimated using a similar formula:
  • 95% confidence interval for proportion ($ ext{p}$):
    ext{p̂} ext{±} 1.96 ext{√}rac{ ext{p̂}(1- ext{p̂})}{n}

Adjusted Confidence Interval

• If sample size n is small, use:
  • Adjusted sample size: n + 4 (add 2 successes and 2 failures).
  • Adjusted sample proportion: ext{p̄} = (x+2)/n^ ext{adjusted}

Example for Proportion

For a sample of 10 cracked tiles out of 125, adjusted sample proportion:

• Adjusted sample size = 129, adjusted proportion = 0.114
• 99% confidence interval:
  0.114 ext{±} 2.575 ext{√}rac{0.114(1-0.114)}{129}

Summary Conclusion

• Understanding confidence intervals is crucial for making inferential statistics about population parameters.
• Different approaches apply based on sample size and distribution characteristics.
• Common pitfalls in interpretation should always be avoided for accurate statistical reporting.