Question

Trial wt % theobromine

1 1.31

2 1.25

3 1.09

4 1.01

5 1.45

Suppose another analysis of theobromine was done on the same chocolate. However, this time a different solvent was used in the extraction procedure. This analysis was performed three times obtaining an average theobromine value of 1.40 wt % with a standard deviation of 0.19. Are the standard deviations of these two data sets able to be pooled? Be sure to indicate the table value that you are using, if applicable.

At the 95% confidence level, can the means of these two methods be considered statistically close enough to be the same? Be sure to indicate the table value that you are using, if applicable.

Answer 1

The two analysis of theobromine are considered to be two independent sample from two independent normal populations.

For the given dataset, mean=1.222 and sd=0.1753

First we perform F test for equal variances. Using MINITAB we get,

Test for Equal Variances

95% Bonferroni confidence intervals for standard deviations

Sample N Lower StDev Upper
1 5 0.234404 0.418688 1.44847
2 3 0.208228 0.435890 3.88649

F-Test (normal distribution)
Test statistic = 0.92, p-value = 0.841

p-value is quite high which indicate the two sample we considered here can assume to be drawn from two population with equal variance. Hence the sample sds can be pooled. Using the pooled sd, we will perform two sample t-test for equality in means. Using MINITAB we get,

Two-Sample T-Test and CI

Sample N Mean StDev SE Mean
1 5 1.222 0.175 0.078
2 3 1.400 0.190 0.11

Difference = mu (1) - mu (2)
Estimate for difference: -0.178000
95% CI for difference: (-0.500250, 0.144250)
T-Test of difference = 0 (vs not =): T-Value = -1.35 P-Value = 0.225 DF = 6
Both use Pooled StDev = 0.1803

Here p-value>0.05. So at the 95% confidence level, the means of these two methods can be considered statistically close enough to be the same.