1 / 19

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.

emarshall
Download Presentation

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

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related