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Inferences on Population Variances: Comparing, Testing, and Confidence Intervals

This chapter introduces the concepts of population variances, sample variances, and their estimators. It explores the sampling distribution of variances, chi-square distributions, and provides methods for constructing confidence intervals and conducting statistical tests on variances.

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Inferences on Population Variances: Comparing, Testing, and Confidence Intervals

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  1. Chapter 7 Inferences Regarding Population Variances

  2. Introduction • Population Variance: Measure of average squared deviation of individual measurements around the mean • Sample Variance: Measure of “average” squared deviation of a sample of measurements around their sample mean. Unbiased estimator of s2

  3. Sampling Distribution of s2 (Normal Data) • Population variance (s2) is a fixed (unknown) parameter based on the population of measurements • Sample variance (s2) varies from sample to sample (just as sample mean does) • When Y~N(m,s), the distribution of (a multiple of) s2 is Chi-Square with n-1 degrees of freedom. (n-1)s2/s2 ~ c2 with df=n-1 • Chi-Square distributions • Positively skewed with positive density over (0,) • Indexed by its degrees of freedom (df) • Mean=df, Variance=2(df) • Critical Values given in Table 7, pp. 1095-1096

  4. Chi-Square Distributions

  5. Chi-Square Distribution Critical Values

  6. Chi-Square Critical Values (2-Sided Tests/CIs) c2L c2U

  7. (1-a)100% Confidence Interval for s2 (or s) • Step 1: Obtain a random sample of n items from the population, and compute s2 • Step 2: Choose confidence level (1-a ) • Step 3: Obtain c2L and c2U from the table of critical values for the chi-square distribution with n-1 df • Step 4: Compute the confidence interval for s2 based on the formula below • Step 5: Obtain confidence interval for standard deviation s by taking square roots of bounds for s2

  8. Statistical Test for s2 • Null and alternative hypotheses • 1-sided (upper tail): H0: s2 s02Ha: s2> s02 • 1-sided (lower tail): H0: s2 s02Ha: s2< s02 • 2-sided: H0: s2= s02Ha: s2 s02 • Test Statistic • Decision Rule based on chi-square distribution w/ df=n-1: • 1-sided (upper tail): Reject H0 if cobs2 > cU2 = ca2 • 1-sided (lower tail): Reject H0 if cobs2 < cL2 = c1-a2 • 2-sided: Reject H0 if cobs2 < cL2 = c1-a/22 (Conclude s2< s02) or if cobs2 > cU2 = ca /22 (Conclude s2> s02)

  9. Inferences Regarding 2 Population Variances • Goal: Compare variances between 2 populations • Parameter: (Ratio is 1 when variances are equal) • Estimator: (Ratio of sample variances) • Distribution of (multiple) of estimator (Normal Data): F-distribution with parameters df1 = n1-1 and df2 = n2-1

  10. Properties of F-Distributions • Take on positive density over the range (0 , ) • Cannot take on negative values • Non-symmetric (skewed right) • Indexed by two degrees of freedom (df1 (numerator df) and df2 (denominator df)) • Critical values given in Table 8, pp 1097-1108 • Parameters of F-distribution:

  11. Critical Values of F-Distributions • Notation: Fa, df1, df2 is the value with upper tail area of a above it for the F-distribution with degrees’ of freedom df1 and df2, respectively • F1-a, df1, df2 = 1/ Fa, df2, df1 (Lower tail critical values can be obtained from upper tail critical values with “reversed” degrees of freedom) • Values given for various values of a, df1, and df2 in Table 8, pp 1097-1108

  12. Test Comparing Two Population Variances • Assumption: the 2 populations are normally distributed

  13. (1-a)100% Confidence Interval for s12/s22 • Obtain ratio of sample variances s12/s22 = (s1/s2)2 • Choose a, and obtain: • FL = F1-a/2, n2-1, n1-1 = 1/ Fa/2, n1-1, n2-1 • FU = Fa/2, n2-1, n1-1 • Compute Confidence Interval: Conclude population variances unequal if interval does not contain 1

  14. Tests Among t ≥ 2 Population Variances • Hartley’s Fmax Test • Must have equal sample sizes (n1 = … = nt) • Test based on assumption of normally distributed data • Uses special table for critical values • Levene’s Test • No assumptions regarding sample sizes/distributions • Uses F-distribution for the test • Bartlett’s Test • Can be used in general situations with grouped data • Test based on assumption of normally distributed data • Uses Chi-square distribution for the test

  15. Hartley’s Fmax Test • H0: s12 = … = st2 (homogeneous variances) • Ha: Population Variances are not all equal • Data: smax2 is largest sample variance, smin2 is smallest • Test Statistic: Fmax = smax2/smin2 • Rejection Region: Fmax F* (Values from class website, indexed by a (.05, .01), t (number of populations) and df2 (n-1, where n is the individual sample sizes)

  16. Levene’s Test • H0: s12 = … = st2 (homogeneous variances) • Ha: Population Variances are not all equal • Data: For each group, obtain the following quantities:

  17. Bartlett’s Test General Test that can be used in many settings with groups • H0: s12 = … = st2 (homogeneous variances) • Ha: Population Variances are not all equal • MSE ≡ Pooled Variance

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