Question

(please include all the steps)

F Distribution Problem The following data lists the annual wages (in thousands of dollars) for randomly selected individuals from three metropolitan areas, Assume the wags are normally distributed and that the samples are independent. Jacksonville, FL Baltimore, MD San Francisco, CA 31.3 64.5 75.9 29.3 45.9 39.8 47.5 46.2 52.3 39.2 45.7 44.9 45.9 59.7 37.5 52.9 34.9 31.5 71.2 74.1 57.5 41.5 49.7 49.8 61.9 65.4 48.1 52.6 47.9 40.1 60.9 58.7 55.9 35.5 42.9 33.4 54.6 39.9 40.9 30.5 50.2 45.4 34.7 a-At α :0.01, is there enough evidence to conclude that the variances in annual San Francisco, CA and Baltimore, MD are different? woges for b- Are the mean annual wages equal for all three cities? Use a 0.10, assume that the population variances are equal.

Answer 1

a.

Test for Equal Variances: Annual wages versus Metropolitan Area

99% Bonferroni confidence intervals for standard deviations

Metropolitan Area N Lower StDev Upper
CA and Baltimore 15 4.11038 6.46470 13.2577
MD 15 4.93303 7.75854 15.9111
San Francisco 13 5.81409 9.41615 20.8342

Bartlett's Test (Normal Distribution)
Test statistic = 1.77, p-value = 0.412

Levene's Test (Any Continuous Distribution)
Test statistic = 0.96, p-value = 0.391

From above tests we see that p-value>0.01 so there is not enough evidence to conclude that the variances in annual wages for three metropolitan areas are different.

b.

Let, A-true mean annual wages for San Francisco; μ2_ true mean annual wages for CA and Baltimore. μ3= true mean annual uages for MD: Null hypothesis. Ho : μ1-μ2-μ3 Alternative hypothesis. HI : At least one μ¡is different from others Let rijannual wages of jth individual selected from ith metropolitan area Then 11 = m Tij sample mean annual wages for San Francisco j-1 60.9692 712 T,Σ xij = sample mean annual wages for CA and Baltimore r. 2- - 45.8667 713 Tij sample mean annual wages for San Francisco 7l 39.1733 7L ΣΣι,-45.0977 rgrand mean- grand mean

SS due to Between metropolitan areasSSetueen - yn,(T, 3423.119 with df 3-1-2 11 Sbetueen-SSbetueen /2 = 3423. 1 19/2 1711 .56 3 i S due total SSatel 5914.91 with df 43 -1 SS due to error SSerrorSStotal- SSbetueen 2491.791 with df-43-3- 40 MSerror SSerror/40 62.2948 F-ratio = MSbetween/M Serror-1711.56/62.2948 = 27.4752 ANOV A table Source ss df MS Betweern 3423.119 2 1711.56 27.4752 Within (E Total 2491.791 40 62.2948 5914.91 42 p-value = P(F > 27.4752F ~ F2.40) = 0.0000 < 0.10 so we reject Ho and conclude that the mean annual wages equal for all three cities