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Chapter 4

Q. 4.20

The %w/w Na_2CO_3 in soda ash can be determined by an acid–base titration. The results obtained by two analysts are shown here. Determine whether the difference in their mean values is significant at α = 0.05.

Analyst A Analyst B
86.82 81.01
87.04 86.15
86.93 81.73
87.01 83.19
86.20 80.27
87.00 83.94

Step-by-Step

Verified Solution

We begin by summarizing the mean and standard deviation for the data reported by each analyst. These values are

\bar{X} _A = 86.83%

s_A = 0.32

\bar{X} _B = 82.71%

s_B = 2.16

A two-tailed F-test of the following null and alternative hypotheses

H_0:     s_A²= s_B²          H_A:     s_A² ≠ s_B²

is used to determine whether a pooled standard deviation can be calculated. The test statistic is

F_{exp}=\frac{s_B^2}{s_A^2} =\frac{(2.16)^2}{(0.32)^2}=45.6

Since F_{exp} is larger than the critical value of 7.15 for F(0.05, 5, 5), the null hypothesis is rejected and the alternative hypothesis that the variances are significantly different is accepted. As a result, a pooled standard deviation cannot be calculated.
The mean values obtained by the two analysts are compared using a twotailed t-test. The null and alternative hypotheses are

H_0:\bar{X}_A=\bar{X}_B        H_A:\bar{X}_A≠\bar{X}_B

Since a pooled standard deviation could not be calculated, the test statistic, t_{exp}, is calculated using equation 4.19

t_{exp}=\frac{\left|\bar{X}_A -\bar{X}_B \right| }{s_{pool}\sqrt{(s_A^2/n_A)+(s_B^2/n_B)} } =\frac{\left|86.83-82.71\right| }{\sqrt{[(0.32)^2/6]+[(2.16)^2/6]} } =4.62

and the degrees of freedom are calculated using equation 4.22

\mathrm{v}=\frac{[(S_A^2/n_A)+(S_B^2/n_B)]^2}{[(S_A^2/n_A)^2/(n_A+1)]+[(S_B^2/n_B)^2/(n_B+1)]} -2      (4.22)

\mathrm{v}=\frac{[(0.32^2/6)+(2.16^2/6)]^2}{[(0.32^2/6)^2/(6+1)]+[(2.16^2/6)/(6+1)]} -2=5.3\simeq 5

The critical value for t(0.05, 5) is 2.57. Since the calculated value of t_exp is greater than t(0.05, 5) we reject the null hypothesis and accept the alternative hypothesis that the mean values for %w/w Na_2CO_3 reported by the two analysts are significantly different at the chosen significance level.