Estimating means, proportions, and variances is crucial for making inferences about populations. This topic covers methods like t-distribution for small samples, normal approximation for proportions, and chi-square distribution for variances.

These techniques help create confidence intervals, giving us a range of likely values for population parameters. Understanding when to use each method and how sample size affects our estimates is key to accurate statistical analysis.

Confidence Intervals for Means

T-Distribution for Small Samples

• Use t-distribution for constructing confidence intervals when population standard deviation unknown and sample size small (typically n < 30)
• Calculate degrees of freedom for t-distribution as n - 1 (n represents sample size)
• Construct confidence interval for population mean using formula $\bar{x} \pm (t (s / \sqrt{n}))$
  • $\bar{x}$ represents sample mean
  • t represents critical value from t-distribution
  • s represents sample standard deviation
  • n represents sample size
• Determine critical value t based on desired confidence level and degrees of freedom
• T-distribution approaches normal distribution as sample size increases
  • Becomes more appropriate for larger samples
• Margin of error in t-distribution confidence interval influenced by:
  • Sample size
  • Sample standard deviation
  • Chosen confidence level
• Example: Constructing 95% confidence interval for mean height of 25 students
  • Sample mean = 170 cm, sample standard deviation = 8 cm
  • Degrees of freedom = 24, t-critical value (95% confidence) = 2.064
  • Confidence interval: 170 ± (2.064 (8 / √25)) = (166.7 cm, 173.3 cm)

Estimating Proportions and Confidence Intervals

Normal Approximation for Proportions

• Use normal approximation to binomial distribution for constructing confidence intervals for population proportions
  • Applicable when np ≥ 5 and n(1-p) ≥ 5
  • n represents sample size
  • p represents sample proportion
• Calculate confidence interval for population proportion using formula $\hat{p} \pm (z \sqrt{(\hat{p}(1-\hat{p}))/n})$
  • $\hat{p}$ represents sample proportion
  • z represents critical value from standard normal distribution
  • n represents sample size
• Compute standard error of proportion as $\sqrt{(\hat{p}(1-\hat{p}))/n}$
• Margin of error in proportion confidence interval influenced by:
  • Sample size
  • Sample proportion
  • Chosen confidence level
• Consider alternative methods for small samples or extreme proportions:
  • Wilson score interval
  • Clopper-Pearson interval
• Normal approximation method assumes sampling distribution of sample proportion approximately normal
  • Generally true for large samples due to Central Limit Theorem
• Example: Estimating proportion of left-handed people in population
  • Sample size = 1000, left-handed individuals = 110
  • Sample proportion = 0.11, z-critical value (95% confidence) = 1.96
  • Confidence interval: 0.11 ± (1.96 * √((0.11 * 0.89) / 1000)) = (0.091, 0.129)

Point Estimates and Confidence Intervals for Variance

Estimating Population Variance and Standard Deviation

• Use sample variance (s²) as point estimate for population variance (σ²)
• Use sample standard deviation (s) to estimate population standard deviation (σ)
• Construct confidence intervals for population variances using chi-square distribution
  • Sampling distribution of sample variance follows chi-square distribution
• Calculate confidence interval for population variance using formula $((n-1)s²)/χ²_{upper} < σ² < ((n-1)s²)/χ²_{lower}$
  • n represents sample size
  • s² represents sample variance
  • χ² represents critical values from chi-square distribution
• Set degrees of freedom for chi-square distribution in variance estimation to n - 1
• Obtain confidence intervals for population standard deviations by taking square root of lower and upper bounds of variance confidence interval
• Width of confidence interval for variances and standard deviations influenced by:
  • Sample size
  • Chosen confidence level
• Assume population normally distributed for validity of confidence intervals
• Example: Estimating population variance of exam scores
  • Sample size = 40, sample variance = 25
  • 95% confidence interval for variance: (17.76, 38.39)
  • 95% confidence interval for standard deviation: (4.21, 6.20)

Central Limit Theorem and Normal Approximations

Applying Central Limit Theorem in Interval Estimation

• Central Limit Theorem (CLT) states sampling distribution of sample mean approaches normal distribution as sample size increases
  • Applies regardless of underlying population distribution
• Apply CLT for practical purposes when sample size ≥ 30
  • Assumes population not extremely skewed
• CLT justifies use of normal approximations in interval estimation for means
  • Applicable even when population distribution unknown or non-normal
• Apply CLT to sampling distribution of sample proportion for proportions
  • Allows normal approximations when np ≥ 5 and n(1-p) ≥ 5
• Standard error of mean (SEM) decreases as sample size increases
  • Key implication of CLT in interval estimation
• CLT enables use of z-scores and t-scores in constructing confidence intervals
  • Based on normal and approximately normal distributions
• Understanding CLT crucial for determining appropriate interval estimation methods
  • Parametric methods for large samples or normal populations
  • Non-parametric or bootstrap methods for small samples or non-normal populations
• Example: Applying CLT to estimate mean household income
  • Sample size = 100, sample mean = $50,000, sample standard deviation = $10,000
  • 95% confidence interval: $50,000 ± (1.96 ($10,000 / √100)) = ($48,040, $51,960)