H-Sec11.2F 2 Variances rev31Jul23 (1)

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

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…

Hypothesis Tests for Two Population Variances, Two-Tail Recent years have seen several national scares involving meat contaminated with Escherichia coli bacteria. The recommended preventative measure is to cook the meat at a required temperature. However, different meat patties cooked for the same amount of time will have different final internal temperatures because of variations in the patties and variations in burner temperatures. A regional hamburger chain will replace its current burners with one of two new digitally controlled models. The chain's purchasing agents have arranged to randomly sample 11 patties of meat cooked by burner model 1 and 13 meat patties cooked by burner model 2 to learn if there is a difference in temperature variation between the two models. If a difference exists, the chain's managers have decided to select the model that provides the smaller variation in final internal meat temperature. Ideally, they would like a test that compares standard deviations, but no such test exists. Instead, they must convert the standard deviations to variances. The managers uses α = 0.10. Formulating the Hypothesis Type of Hypothesis Test: Researching a Hypothesis. Build the translation into the Alterna Hypotheses: H0: σ1² = σ2² HA: σ1² ≠ σ2² Type of Tail: Two-Tail Test 2 tail test Data α = 0.02 2 tail test Model 1 Model 2 Keep in mind that Model 1 may not be Population 1 180.0 178.6 And that Model 2 may not be Population 2 181.5 182.3 178.9 177.5 Since the hypothesis is two-tailed: 176.4 180.6 Before assigning data sets to populations, com 180.7 178.3 The larger sample variance becomes Populatio 181.0 180.7 The smaller sample variance becomes Populati 180.3 181.4 184.6 180.5 185.6 179.6 179.7 178.2 178.9 182.0 181.5 180.8 Translate English: learn if there is a difference in temperature variation between the two models To Statistics: HA: σ1² ≠ σ2² The F test statistic is calculated from the larger

2 tail test Calculations Model 1 Model 2 Remember these are variances not standard 122500 62500 Because Model 1 variance is greater than Model 2 varia TEST STAT 1.9600 we can continue with the labels since the number =VAR.S(C31:C43) =VAR.S(D31:D43) =MAX(C47:D47)/MIN(C47:D47) Check that your sample sizes are in the s Model 1 Model 2 n = 20 20 =COUNT(C31:C43) =COUNT(D31:D43) d.f. = 19 19 =C54-1 =D54-1 α/2 = 0.01 (This is a two-tail situation) =C28/2 3.027 =IF(C47>D47,F.INV.RT(C56,C55,D55),F.INV.RT(C56,D55,C55)) There is only 1 critical value here for the 2 tail test. Decision Rule 2 tail test If 2.71 > 2.75, then reject H0. Otherwise, do not reject H0. Conclusion Do not reject H0. Repeat for p-value Calculations F = 1.96 =C48 d.f. = 19 19 =C55 =D55 p-value = 0.151474 =IF(C47>D47,F.DIST.RT(C70,C71,D71)*2,F.DIST.RT(C70,D α = 0.02 =C28 If p-value < α, reject H0. Otherwise, do not reject H0. Decision Rule If p-value < α, reject H0. Otherwise, do not reject H0. If 0.104 < 0.100, reject H0. Otherwise, do not reject H0. Conclusion Do not reject H0. Which model will they buy? Why? IMPORTANT : The first column should contain the la s² = F = s² = F = F -critical = If F > F -critical, then reject H0. Otherwise, do not reject H0. There is insufficient evidence to support the claim that there is a difference in variation of internal meat temperatures. There is insufficient evidence to support the claim that there is a difference in variation of internal meat temperatures.

Item Sample 1 Sample 2 1 3.8 9.5 2 3.2 5.9 3 4.4 10.4 ative Hypothesis 4 4.4 6.6 5 5.5 8.5 6 3.4 5.7 7 5.3 8.9 8 4.9 3.7 9 2.4 8.3 10 6.9 8.4 1.701778 4.198778 mpute variances. on 1. tion 2. r variance over the smaller variance.

d deviations ance, ring matches. same order as your sample standard deviations. D71,C71)*2) arger variance

Hypothesis Tests for Two Population Variances, Two-Tail Recent years have seen several national scares involving meat contaminated with Escherichia coli bacteria. The recommended preventative measure is to cook the meat at a required temperature. However, different meat patties cooked for the same amount of time will have different final internal temperatures because of variations in the patties and variations in burner temperatures. A regional hamburger chain will replace its current burners with one of two new digitally controlled models. The chain's purchasing agents have arranged to randomly sample 11 patties of meat cooked by burner model 1 and 13 meat patties cooked by burner model 2 to learn if there is a difference in temperature variation between the two models. If a difference exists, the chain's managers have decided to select the model that provides the smaller variation in final internal meat temperature. Ideally, they would like a test that compares standard deviations, but no such test exists. Instead, they must convert the standard deviations to variances. The managers uses α = 0.10. Formulating the Hypothesis Type of Hypothesis Test: Researching a Hypothesis. Build the translation into the Alternative Hypoth Hypotheses: H0: σ1² = σ2² HA: σ1² ≠ σ2² Type of Tail: Two-Tail Test Data α = 0.1 Model 1 Model 2 Keep in mind that Model 1 may not be Population 1 180.0 178.6 And that Model 2 may not be Population 2 181.5 182.3 178.9 177.5 Since the hypothesis is two-tailed: 176.4 180.6 Before assigning data sets to populations, compute varian 180.7 178.3 The larger sample variance becomes Population 1. 181.0 180.7 The smaller sample variance becomes Population 2. 180.3 181.4 184.6 180.5 185.6 179.6 179.7 178.2 178.9 182.0 181.5 180.8 Calculations Translate English: learn if there is a difference in temperature variation between the two models To Statistics: HA: σ1² ≠ σ2² The F test statistic is calculated from the larger variance o

Model 1 Model 2 6.656909 2.452692 Because Model 1 variance is greater than Model 2 variance, test stat 2.714123 we can continue with the labels since the numbering matche =VAR.S(C31:C43) =VAR.S(D31:D43) =MAX(C47:D47)/MIN(C47:D47) Model 1 Model 2 n = 11 13 =COUNT(C31:C43) =COUNT(D31:D43) d.f. = 10 12 =C54-1 =D54-1 α/2 = 0.05 (This is a two-tail situation) =C28/2 2.753387 =IF(C47>D47,F.INV.RT(C56,C55,D55),F.INV.RT(C56,D55,C55)) There is only 1 critical value here for the 2 tail test. Decision Rule If 2.71 > 2.75, then reject H0. Otherwise, do not reject H0. Conclusion Do not reject H0. Repeat for p-value Calculations F = 2.714123 =C48 d.f. = 10 12 =C55 =D55 p-value = 0.104728 =IF(C47>D47,F.DIST.RT(C70,C71,D71)*2,F.DIST.RT(C70,D71,C71)*2) α = 0.1 =C28 Decision Rule If p-value < α, reject H0. Otherwise, do not reject H0. If 0.104 < 0.100, reject H0. Otherwise, do not reject H0. Conclusion Do not reject H0. Which model will they buy? Why? Uncertain. They will have to use a criteria other than variance because the variances are not differe s² = F = s² = F = F -critical = If F > F -critical, then reject H0. Otherwise, do not reject H0. There is insufficient evidence to support the claim that there is a difference in variation of internal meat temperatures. There is insufficient evidence to support the claim that there is a difference in variation of internal meat temperatures.

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Hypothesis Tests for Two Population Variances, One Tail Consider the systems development group for a Midwestern bank, which has developed a new soft algorithm for its automatic teller machines (ATMs). Although reducing average transaction time is objective, the systems programmers also want to reduce the variability in transaction speed. They the standard deviation for transaction time will be less with the new software than it was with the algorithm. For their analysis, the programmers have performed 7 test runs using the original softw 11 test runs using the new system. Although the managers want to determine the standard deviati transaction time, they must perform the test as a test of variances because no method exists for te 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 Data α = 0.01 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 Calculations Old (Pop 1) New (Pop 2) 612.6762 51.4940 =VAR.S(C28:C38) 11.8980 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 = F = ß Because this is a one tail test, the Alternate Hypot population 2. F must be calculated that way even if th

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. Check First cells in both ranges contain label on Select Upper-tail Chose a Title , if desired Click OK

ftware an believe e old ware and tion of esting 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. software than it was with the old ted from the larger hypothesized variance over the smaller hypothesized variance. thesis informs us that we expect population 1 to be larger than he calculated values for s² turn out otherwise