A study was performed to compare the variability of sulfur content of gasoline w

Question

A study was performed to compare the variability of sulfur content of gasoline we buy from two producers Producer 1, & Producer 2). Analysis of 16 samples (bottles) from each producer shows that standard deviations are 1.52 and 1.38 for producers 1 and 2 respectively. Use =2%

A: Test the hypothesis that there is no difference in the variances of the pr oduct we purchase from the two producers.

B: suppose in real case, the standard deviation of producer 1 is 2.76. what is the probability of type 2 error. Determine power of the test

C. If in part b, you want to ha ve a power =0.9, what (equal) sample size we should use

Explanation / Answer

Test and CI for Two Variances

Method

Null hypothesis         (First) / (Second) = 1
Alternative hypothesis (First) / (Second) 1
Significance level      = 0.02

F method was used. This method is accurate for normal data only.

Statistics

                               98% CI for
Sample   N StDev Variance      StDevs
First   16 1.520     2.310 (1.065, 2.574)
Second 16 1.380     1.904 (0.967, 2.337)

Ratio of standard deviations = 1.101
Ratio of variances = 1.213

98% Confidence Intervals

                            CI for
         CI for StDev      Variance
Method       Ratio           Ratio
F       (0.587, 2.067) (0.344, 4.273)

Tests

                       Test
Method DF1 DF2 Statistic P-Value
F       15   15       1.21      0.713

p-value = 0.713 > 0.05

hence we fail to reject the null hypothesis

b)

Power and Sample Size

Test for Two Standard Deviations

Testing (StDev 1 / StDev 2) = 1 (versus )
Calculating power for (StDev 1 / StDev 2) = ratio
= 0.02
Method: Levene’s Test

       Sample
Ratio    Size     Power
    2      16     0.464335

c)

Power and Sample Size

Test for Two Standard Deviations

Testing (StDev 1 / StDev 2) = 1 (versus )
Calculating power for (StDev 1 / StDev 2) = ratio
= 0.02
Method: Levene’s Test

       Sample Target
Ratio    Size   Power Actual Power
    2        39     0.9      0.906619

The sample size is for each group.

n1 =n2 = 39

b)

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