Question 4.16: Before determining the amount of Na2CO3 in an unknown sample...
Before determining the amount of \mathrm{Na}_{2} \mathrm{CO}_{3} in an unknown sample, a student decides to check her procedure by analyzing a sample known to contain 98.76 \% \mathrm{w} / \mathrm{w}~ \mathrm{Na}_{2} \mathrm{CO}_{3}. Five replicate determinations of the %w/w \mathrm{Na}_{2} \mathrm{CO}_{3} in the standard were made with the following results
98.71 \% \quad 98.59 \% \quad 98.62 \% \quad 98.44 \% \quad 98.58 \%
Is the mean for these five trials significantly different from the accepted value at the 95 \% confidence level (\alpha=0.05) ?
The mean and standard deviation for the five trials are
\bar{X}=98.59 \quad s=0.0973
Since there is no reason to believe that \bar{X} must be either larger or smaller than \mu, the use of a two-tailed significance test is appropriate. The null and alternative hypotheses are
H_{0}: \quad \bar{X}=\mu \quad H_{\mathrm{A}}: \quad \bar{X} \neq \mu
The test statistic is
t_{\exp }=\frac{|\mu-\bar{X}| \times \sqrt{n}}{s}=\frac{|98.76-98.59| \times \sqrt{5}}{0.0973}=3.91
The critical value for t(0.05,4), as found in Appendix 1 \mathrm{~B}, is 2.78 . Since t_{\exp } is greater than t(0.05,4), we must reject the null hypothesis and accept the alternative hypothesis. At the 95\% confidence level the difference between \bar{X} and \mu is significant and cannot be explained by indeterminate sources of error. There is evidence, therefore, that the results are affected by a determinate source of error.
| Appendix 1B t-Table^a |
||||
| Value of t for confidence interval of : Critical value of |t| for α values of : Degrees of Freedom |
90% 0.10 |
95 % 0.05 |
98 % 0.02 |
99 % 0.01 |
| 1 | 6.31 | 12.71 | 31.82 | 63.66 |
| 2 | 2.92 | 4.30 | 6.96 | 9.92 |
| 3 | 2.35 | 3.18 | 4.54 | 5.84 |
| 4 | 2.13 | 2.78 | 3.75 | 4.60 |
| 5 | 2.02 | 2.57 | 3.36 | 4.03 |
| 6 | 1.94 | 2.45 | 3.14 | 3.71 |
| 7 | 1.89 | 2.36 | 3.00 | 3.50 |
| 8 | 1.86 | 2.31 | 2.90 | 3.36 |
| 9 | 1.83 | 2.26 | 2.82 | 3.25 |
| 10 | 1.81 | 2.23 | 2.76 | 3.17 |
| 12 | 1.78 | 2.18 | 2.68 | 3.05 |
| 14 | 1.76 | 2.14 | 2.62 | 2.98 |
| 16 | 1.75 | 2.12 | 2.58 | 2.92 |
| 18 | 1.73 | 2.10 | 2.55 | 2.88 |
| 20 | 1.72 | 2.09 | 2.53 | 2.85 |
| 30 | 1.70 | 2.04 | 2.46 | 2.75 |
| 50 | 1.68 | 2.01 | 2.40 | 2.68 |
| \infty | 1.64 | 1.96 | 2.33 | 2.58 |
| ^aThe t-values in this table are for a two-tailed test. For a one-tailed test, the α values for each column are half of the stated value. For example, the first column for a one-tailed test is for the 95% confidence level, α = 0.05. |
||||