( ) 2 ( ) 2 where s 1 > s 2

Size: px

Start display at page:

Download "( ) 2 ( ) 2 where s 1 > s 2"

Marybeth Blair
6 years ago
Views:

1 Section 9 4: Testing a Claim about the Difference in 2 Population Standard Deviations Test H 0 : σ 1 =σ 2 there is no difference in Population Standard Deviations σ 1 σ 2 = 0 against H 1 : σ 1 >σ 2 or H 1 : σ 1 σ 2 where s 1 > at a significance level of α 1. Both populations are normal. Requirements 2. An independent random sample of each population is taken. The sample with the larger sample standard deviation will be labeled Sample 1. The sample with the smaller sample standard deviation will be labeled Sample 2. Notation for the Samples of Two Population Standard Deviations The sample with the larger sample The sample with the smaller sample σ 1 = Population Standard Deviation σ 2 = Population Standard Deviation n 1 = sample size n 2 = sample size s 1 = Sample Standard Deviation = Sample Standard Deviation from Population 1 from Population 2 Testing a Claim about the Difference in 2 Population Standard Deviations with H 0 : σ 1 =σ 2 Test Statistic: F = s 1 where s 1 > with the Numerators DF = n 1 1 and the Denominators DF = n 2 1 Note: By selecting s 1 > we can always use a right tail test. Find the right tail critical value of the F distribution and compare it against the F test statistic. If the F test statistic is greater than the right tail critical value of the F distribution then reject H 0 : σ 1 =σ 2 If the F test statistic is less than the right tail critical value of the F distribution then do not reject H 0 : σ 1 =σ 2 Section 9 4 Lecture Page 1 of Eitel

2 Testing a Claim about the Difference in 2 Population Standard Deviations Example 1 (Right Tail Test) A study of the amount of variance in the amount of caffeine in Bayer Aspirin Excedrin was conducted by a local hospital. A random sample of 41 tablets of 500 mg. Bayer Aspirin found that they contained an average of 20 mg. of caffeine and had a standard deviation of.32 mg of caffeine. A random sample of 61 tablets of 500 mg. of Excedrin found that they contained an average of 20 mg. of caffeine had a standard deviation of.29 mg. of caffeine. Use a α =.01 significant level to test the claim that 500 mg. tablets of Bayer Aspirin have a larger standard deviation in caffeine than 500 mg. tablets of Excedrin. The sample with the larger sample The sample with the smaller sample Bayer Aspirin Excedrin s 1 =.32 mg. =.29 mg. n 1 = 41 Numerator DF = 40 n 2 = 61 Denominator DF = 60 Ho: σ 1 = σ 2 H1: σ 1 > σ 2 Right Tail Test of H 0 : σ 1 = σ 2 α =.01 Numerator DXF = 40 Denominator DXF = 60 Test Statistic: F = s 1 where s 1 > Do Not F = α =.01 F F = F = Conclusion based on H 0 : Do not Conclusion based on the problem: There is not sufficient evidence at the α =.01 level to reject the hypothesis that 500 mg. tablets of Bayer Aspirin have the same standard deviation in the amount of caffeine as 500 mg. Excedrin tablets. Section 9 4 Lecture Page 2 of Eitel

3 F Distribution: Critical Values for a Right Tail with Area.01 DXF Inf. DXF Section 9 4 Lecture Page 3 of Eitel

4 Example 2 (Left Tail Question converted into a Right Tail Test) A local weather station reported stated both Redding and Sacramento have an average yearly rainfall of 28 inches. A viewer from Redding called in and said that it seemed like Redding had a lot more variation in rainfall from year to year than Sacramento did. A random sample of 61 years of rainfall in Redding had an average yearly rainfall of 28 inches a year with a sample standard deviation of 4 inches a year. A random sample of 121 years of rainfall in Sacramento had an average yearly rainfall of 28 inches a year with a sample standard deviation of 6 inches a year. Use a α =.05 significant level to test the claim that Redding has a lower standard deviation in yearly rainfall than Sacramento. The sample with the larger sample The sample with the smaller sample Sacramento Redding s 1 = 6 in. = 4 in. n 1 =121 Numerator DF = 120 n 2 = 61 Denominator DF = 60 Note: If =1 (Redding) is lower than s 1 = 6 (Sacramento) then H1 could be σ 2 < σ 1 but we want a right tail test so we test H1 as σ 1 > σ 2 Ho: σ 1 = σ 2 H1: σ 1 > σ 2 Graph of Critical information: α =.05 Numerator DXF = 120 Denominator DXF = 60 Test Statistic: F = s 1 where s 1 > Do Not F = α =.05 F F = F = 2.25 Conclusion based on H 0 : Conclusion based on the problem: There is sufficient evidence at the α =.05 level to reject the hypothesis that Sacramento and Redding have the same standard deviation in annual rainfall. Sacramento's yearly rainfall varies more than Redding's yearly rainfall. Section 9 4 Lecture Page 4 of Eitel

5 F Distribution: Critical Values for a Right Tail with Area.05 DXF Inf. DXF Section 9 4 Lecture Page 5 of Eitel

6 Testing a Claim about the Difference in 2 Population Standard Deviations Example 3 (Two Tail Test) A quality control study of the daly production of 1% milk at the local bottling plant was made. A sample of 41 gallon containers of 1% milk was taken from Production Line A showed that the average grams of fat wa.3 grams with a standard deviation of.92 grams. A sample of 25 gallon containers of 1% milk was taken from Production Line B showed that the average grams of fat wa.4 grams with a standard deviation of.52 grams. Assume that both populations are normally distributed. Use a α =.02 significant level to test the claim that the two populations have different Standard Deviations. The sample with the larger The sample with the smaller Production Line A Production Line B s 1 =.92 =.52 n 1 = 41 Numerator DF = 40 n 2 = 25 Denominator DF = 24 Ho: σ 1 = σ 2 H1: σ 1 σ 2 Graph of Critical information: α =.02 so α 2 =.01 Numerator DXF = 40 Denominator DXF = 24 Test Statistic: F = s 1 where s 1 > Do Not F = α 2 =.01 F F = F = Conclusion based on H 0 : Conclusion based on the problem: There is sufficient evidence at the α =.02 level to support the claim that that production Line A and production Line B population have different standard deviations. Section 9 4 Lecture Page 6 of Eitel

7 F Distribution: Critical Values for a Right Tail with Area.01 DXF Inf. DXF Section 9 4 Lecture Page 7 of Eitel

( ) 2 ( ) 2 where s 1 > s 2

( ) 2 ( ) 2 where s 1 > s 2 Section 9 3: Testing a Claim about the Difference in! 2 Population Standard Deviations Test H 0 : σ 1 = σ 2 there is no difference in Population Standard Deviations σ 1 σ 2 = 0 against H 1 : σ 1 > σ 2