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Ch 8 - Confidence Intervals in Business Statistics

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Terms in this set (21)

  • What is a confidence interval?
    A confidence interval is a range of values within which a population parameter is expected to lie with a specified probability.
  • Define point estimate.
    A point estimate is a single value, such as a sample mean, used as the best estimate of an unknown population parameter.
  • What is the margin of error (ME) in confidence intervals?
    The margin of error is the amount added and subtracted from the point estimate to form the confidence interval.
  • Explain the empirical rule for normal distributions.
    Approximately 68% of values fall within ±1 standard deviation, 95% within ±2, and 99.7% within ±3 standard deviations from the mean.
  • What does the Central Limit Theorem state?
    For large samples (n ≥ 30), the sampling distribution of the sample mean is approximately normal regardless of the population distribution.
  • What is the critical value in confidence intervals?
    The critical value (zα/2 or tα/2) determines the width of the confidence interval for a given confidence level.
  • Common z-scores for 90%, 95%, and 99% confidence levels?
    90%: 1.645, 95%: 1.96, 99%: 2.575
  • When to use z-distribution for confidence intervals?
    Use the z-distribution when population standard deviation (σ) is known and sample size is large (n ≥ 30) or population is normal.
  • Formula for confidence interval when σ is known?
    Confidence interval = sample mean ± critical z-score × (σ / √n).
  • What is the finite population correction (FPC)?
    FPC adjusts the standard error when the sample size is a significant fraction of the population (n/N > 0.05), calculated as \(\sqrt{\frac{N-n}{N-1}}\).
  • When to use t-distribution for confidence intervals?
    Use the t-distribution when σ is unknown and the population is normally distributed, replacing σ with sample standard deviation s.
  • Formula for confidence interval when σ is unknown?
    Confidence interval = sample mean ± critical t-score × (s / √n), with degrees of freedom df = n - 1.
  • Conditions for approximating binomial distribution with normal distribution?
    Sample size must satisfy np ≥ 5 and n(1-p) ≥ 5.
  • Formula for confidence interval for a population proportion?
    Confidence interval = sample proportion ± critical z-score × √(p(1-p)/n).
  • When to apply finite population correction for proportions?
    Apply FPC when the sample is a significant fraction of the population, adjusting the standard error accordingly.
  • What affects the width of a confidence interval?
    The confidence level, sample size, and variability in the data affect the width of the confidence interval.
  • Summary: When to use z vs t distributions?
    Use z-distribution when σ is known; use t-distribution when σ is unknown and population is normal.
  • What is the significance level (α) in confidence intervals?
    The significance level α is the probability that the confidence interval does not contain the true population parameter, equal to 1 minus the confidence level.
  • Example: Calculate 90% CI for mean with σ known, n=35, mean=145.5, σ=30.2.
    Standard error = 30.2/√35 ≈ 5.105; critical z = 1.645; CI = 145.5 ± 1.645 × 5.105 = (137.10, 153.90).
  • Example: Calculate 95% CI for mean with σ unknown, n=15, mean=5.11, s=0.85.
    Standard error = 0.85/√15 ≈ 0.2195; critical t (df=14) = 2.145; CI = 5.11 ± 2.145 × 0.2195 = (4.64, 5.58).
  • Example: Calculate 98% CI for proportion with n=100, successes=22.
    Sample proportion p = 0.22; standard error = √(0.22×0.78/100) = 0.041; critical z = 2.33; CI = 0.22 ± 2.33 × 0.041 = (0.124, 0.316).