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Chapter 4: Sampling and Statistical Inference

Chapter 4: Sampling and Statistical Inference. Part 2: Estimation. Types of Estimates. Point estimate – a single number used to estimate a population parameter Interval estimate – a range of values between which a population parameter is believed to be. Common Point Estimates.

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Chapter 4: Sampling and Statistical Inference

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  1. Chapter 4: Sampling and Statistical Inference Part 2: Estimation

  2. Types of Estimates • Point estimate – a single number used to estimate a population parameter • Interval estimate – a range of values between which a population parameter is believed to be

  3. Common Point Estimates

  4. Theoretical Issues • Unbiased estimator – one for which the expected value equals the population parameter it is intended to estimate • The sample variance is an unbiased estimator for the population variance

  5. Confidence Intervals • Confidence interval (CI) – an interval estimated that specifies the likelihood that the interval contains the true population parameter • Level of confidence (1 – a) – the probability that the CI contains the true population parameter, usually expressed as a percentage (90%, 95%, 99% are most common).

  6. Confidence Intervals for the Mean - Rationale

  7. Confidence Interval for the Mean – s Known A 100(1 – a)% CI is: x  z/2(/n) z/2 may be found from Table A.1 or using the Excel function NORMSINV(1-a/2)

  8. Confidence Interval for the Mean, s Unknown • A 100(1 – a)% CI is: x  t/2,n-1(s/n) t/2,n-1is the value from a t-distribution with n-1 degrees of freedom, from Table A.3 or the Excel function TINV(a, n-1)

  9. Relationship Between Normal Distribution and t-distribution The t-distribution yields larger confidence intervals for smaller sample sizes.

  10. PHStat Tool: Confidence Intervals for the Mean • PHStat menu > Confidence Intervals > Estimate for the mean, sigma known…, or Estimate for the mean, sigma unknown…

  11. PHStat Tool: Confidence Intervals for the Mean - Dialog Enter the confidence level Choose specification of sample statistics Check Finite Population Correction box if appropriate

  12. PHStat Tool: Confidence Intervals for the Mean - Results

  13. Confidence Intervals for Proportions • Sample proportion: p = x/n • x = number in sample having desired characteristic • n = sample size • The sampling distribution of p has mean p and variance p(1 – p)/n • When np and n(1 – p) are at least 5, the sampling distribution of p approach a normal distribution

  14. Confidence Intervals for Proportions • A 100(1 – a)% CI is: PHStat tool is available under Confidence Intervals option

  15. Confidence Intervals and Sample Size • CI for the mean, s known • Sample size needed for half-width of at most E is n  (z/2)2(s2)/E2 • CI for a proportion • Sample size needed for half-width of at most E is • Use p as an estimate of p or 0.5 for the most conservative estimate

  16. PHStat Tool: Sample Size Determination • PHStat menu > Sample Size > Determination for the Mean or Determination for the Proportion Enter s, E, and confidence level Check Finite Population Correction box if appropriate

  17. Confidence Intervals for Population Total • A 100(1 – a)% CI is: N x  tn-1,a/2 PHStat tool is available under Confidence Intervals option

  18. Population 1 Population 2 Mean m1 m2 Standard deviation s1 s2 Point estimate x1 x2 Sample size n1 n2 Confidence Intervals for Differences Between Means Point estimate for the difference in means, m1 – m2, is given by x1 - x2

  19. Independent Samples With Unequal Variances • A 100(1 – a)% CI is: x1 -x2 (ta/2, df*) Fractional values rounded down df* =

  20. Independent Samples With Equal Variances • A 100(1 – a)% CI is: x1 -x2 (ta/2, n1 + n2 – 2) where sp is a common “pooled” standard deviation. Must assume the variances of the two populations are equal.

  21. Paired Samples • A 100(1 – a)% CI is: D  (tn-1,a/2) sD/n Di = difference for each pair of observations D = average of differences PHStat tool available in the Confidence Intervals menu

  22. Differences Between Proportions • A 100(1 – a)% CI is: Applies when nipi and ni(1 – pi) are greater than 5

  23. Sampling Distribution of s • The sample standard deviation, s, is a point estimate for the population standard deviation, s • The sampling distribution of s has a chi-square (c2) distribution with n-1 df • See Table A.4 • CHIDIST(x, deg_freedom) returns probability to the right of x • CHIINV(probability, deg_freedom) returns the value of x for a specified right-tail probability

  24. Confidence Intervals for the Variance • A 100(1 – a)% CI is:

  25. PHStat Tool: Confidence Intervals for Variance - Dialog • PHStat menu > Confidence Intervals > Estimate for the Population Variance Enter sample size, standard deviation, and confidence level

  26. PHStat Tool: Confidence Intervals for Variance - Results

  27. Time Series Data • Confidence intervals only make sense for stationary time series data

  28. Probability Intervals • A 100(1 – a)% probability interval for a random variable X is an interval [A,B] such that P(A X  B) = 1 – a • Do not confuse a confidence interval with a probability interval; confidence intervals are probability intervals for sampling distributions, not for the distribution of the random variable.

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