Before determining the amount of Na2CO3 in an unknown sample, a student decides to check

Question

Before determining the amount of [latex]\mathrm{Na}_{2} \mathrm{CO}_{3}[/latex] in an unknown sample, a student decides to check her procedure by analyzing a sample known to contain [latex]98.76 \% \mathrm{w} / \mathrm{w}~ \mathrm{Na}_{2} \mathrm{CO}_{3}[/latex]. Five replicate determinations of the %w/w [latex]\mathrm{Na}_{2} \mathrm{CO}_{3}[/latex] in the standard were made with the following results

[latex]98.71 \% \quad 98.59 \% \quad 98.62 \% \quad 98.44 \% \quad 98.58 \%[/latex]

Is the mean for these five trials significantly different from the accepted value at the [latex]95 \%[/latex] confidence level [latex](\alpha=0.05)[/latex] ?

Accepted Answer

The mean and standard deviation for the five trials are

[latex]\bar{X}=98.59 \quad s=0.0973[/latex]

Since there is no reason to believe that [latex]\bar{X}[/latex] must be either larger or smaller than [latex]\mu[/latex], the use of a two-tailed significance test is appropriate. The null and alternative hypotheses are

[latex]H_{0}: \quad \bar{X}=\mu \quad H_{\mathrm{A}}: \quad \bar{X} eq \mu[/latex]

The test statistic is

[latex]t_{\exp }=\frac{|\mu-\bar{X}| imes \sqrt{n}}{s}=\frac{|98.76-98.59| imes \sqrt{5}}{0.0973}=3.91[/latex]

The critical value for [latex]t(0.05,4)[/latex], as found in Appendix [latex]1 \mathrm{~B}[/latex], is 2.78 . Since [latex]t_{\exp }[/latex] is greater than [latex]t(0.05,4)[/latex], we must reject the null hypothesis and accept the alternative hypothesis. At the 95\% confidence level the difference between [latex]\bar{X}[/latex] and [latex]\mu[/latex] 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.

Question 4.16: Before determining the amount of Na2CO3 in an unknown sample...

Related Answered Questions

A manufacturer’s process for analyzing aspirin tablets has a known variance of 25. A ...

Verified Answer:

The amount of aspirin in the analgesic tablets from a particular manufacturer is known to ...

Verified Answer:

Tables 4.1 and 4.8 show results for two separate experiments to determine the mass of a ...

Verified Answer:

What is the 95% confidence interval for the data in Table 4.1? ...

Verified Answer:

The following masses, in grams, were recorded in an experiment to determine the average mass of a U.S. penny. 3.067 3.049 3.039 2.514 3.048 3.079 3.094 3.109 3.102 Determine if the value of 2.514 g is an outlier at α = 0.05. ...

Verified Answer:

Marecek and colleagues developed a new electrochemical method for the rapid quantitative analysis of the antibiotic monensin in the fermentation vats used during its production.^9 The standard method for the analysis, which is based on a test for microbiological activity, is both difficult and ...

Verified Answer:

The %w/w Na2CO3 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. ...

Verified Answer:

Tables 4.1 and 4.8 show results for two separate experiments to determine the mass of a circulating U.S. penny. Determine whether there is a difference in the precisions of these analyses at α = 0.05. ...

Verified Answer:

The quantity of charge, Q, in coulombs passing through an electrical circuit is Q = I × t where I is the current in amperes and t is the time in seconds. When a current of 0.15 ± 0.01 A passes through the circuit for 120 ± 1 s, the total charge is Q = (0.15 A) × (120 s) = 18 C Calculate the ...

Verified Answer:

The class A 10-mL pipet characterized in Table 4.8 is used to deliver two successive volumes. Calculate the absolute and relative uncertainties for the total delivered volume. ...

Verified Answer: