H-Sec11

Hypothesis Tests for Two Population Variances, One Tail Consider the systems development group for a Midwestern bank, which has developed a new software algorithm for its automatic teller machines (ATMs). Although reducing average transaction time is an objective, the systems programmers also want to reduce the variability in transaction speed. They belie the standard deviation for transaction time will be less with the new software than it was with the old algorithm. For their analysis, the programmers have performed 7 test runs using the original software a 11 test runs using the new system. Although the managers want to determine the standard deviation o transaction time, they must perform the test as a test of variances because no method exists for testing standard deviations directly. α = 0.01. Formulating the Hypothesis Type of Hypothesis Test: Researching a Hypothesis. Build the translation into the Altern To Statistics: HA: σ_New_Software < σ_Old_Algorithm Type of Tail: One-Tail Hypotheses: Any One-Tail Hypothesis Test for Two Population Varia HA: σ1² > σ2² The next step is determining which Population (1 or 2 in this case New Software or Old Algorithm. The New Software is Population 2 The Old Algorithm is Population 1 1 tail test Data α = 0.025 Old Algorithm New Software Keep in mind that the Old Algorithm 38.9 22.8 And that the New Software may not b 23.2 20.0 49.2 26.5 Since the hypothesis is one-tailed: 66.8 37.9 Before assigning data sets to 65.5 27.2 The larger variance in the hy 74.6 39.6 The smaller variance in the h 7.7 34.1 39.4 20.9 30.3 29.1 612.7 51.5 var s Calculations Old (Pop 1) New (Pop 2) 1361.0000 1184.0000 =VAR.S(C28:C38) Translate English: the standard deviation for transaction time will be less with the new s algorithm H0: σ1² ≤ σ2² The F test statistic is calculat s² =

TEST STAT 1.1495 =C42/D42 Old (Pop 1) New (Pop 2) Check that your sa n = 13 23 =COUNT(C28:C38) df = 12 22 =C47-1 2.6017 =F.INV.RT(D25,C48,D48) Decision Rule ^ Right Tail Test We made it a right tail when we u If 11.90 > 5.39, then reject H0. Otherwise, do not reject H0. Conclusion Reject H0. Repeat of the process using Data Analysis F-Test Two-Sample f Original Softwar New System 38.9 22.8 23.2 20.0 Mean 49.2 26.5 Variance 66.8 37.9 Observations 65.5 27.2 df 74.6 39.6 F 7.7 34.1 P(F<=f) one-tail 39.4 F Critical one-tail 20.9 30.3 29.1 Repeat of the process using PHStat F = ß Because this is a one tail test, the Alternate Hypot population 2. F must be calculated that way even if th F- critical = ß This follows the same pattern as in the prior note If F > F -critical, then reject H0. Otherwise, do not reject H0. There is sufficient evidence to conclude that the new software produces shorter variances in transaction time than the old algorithm. Select DATA > Data Analysis > F-Test Two Sample for Variances > OK Select the range of cells for Input / Variable 1 Range including header Select the range of cells for Input / Variable 2 Range including header Check Labels on Input Alpha of 0.01 Chose the Output Option of your choice Click OK Since F is greater tha Select PHStat > Two-Sample Tests (Unsummarized Data) > F Test for Differences in Two

Do not use PHStat for Upper-Tail Test. PHStat uses the Two-Tail Test technique for both Two-Tail and Upper-Tail Test. This means that if the hyopthesized variance and the computed variance are both the larger variance, then PHStat's technique will work. Input the Level of Significance as 0.01 Select the range of cells for Population 1 Sample Cell Range: including header Select the range of cells for Population 2 Sample Cell Range: including header Check First cells in both ranges contain label on Select Upper-tail Chose a Title , if desired Click OK

e eve and of g native Hypothesis ances will look like this. 2) belong to which concept, may not be Population 1 be Population 2 o populations, compare hypothesized variances. ypotheses becomes Population 1. hypotheses becomes Population 2. 1 tail test software than it was with the old ted from the larger hypothesized variance over the smaller hypothesized variance.

1 tail test ample sizes are in the same order as your sample standard deviations. used the larger variance in the first column above! P Value Right Tail 0.3717 p value for Variances Original Software New System 46.5571428571429 29.8 612.676190476191 51.494 7 11 6 10 11.8980112338562 0.000473971575695 5.3858110448458 thesis informs us that we expect population 1 to be larger than he calculated values for s² turn out otherwise an F Critical one-tail , there is sufficicent evidence to reject the null hypothesis o Variances…