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## 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

## 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.