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

Question

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 means of these analyses at [latex]\alpha=0.05[/latex].

Accepted Answer

To begin with, we must determine whether the variances for the two analyses are significantly different. This is done using an F-test as outlined in Example 4.18. Since no significant difference was found, a pooled standard deviation with 10 degrees of freedom is calculated

[latex] \begin{aligned} s_{ ext {pool }} & =\sqrt{\frac{\left(n_{\mathrm{A}}-1 ight) s_{\mathrm{A}}^{2}+\left(n_{\mathrm{B}}-1 ight) s_{\mathrm{B}}^{2}}{n_{\mathrm{A}}+n_{\mathrm{B}}-2}} \ & =\sqrt{\frac{(7-1)(0.00259)+(5-1)(0.00138)}{7+5-2}} \ & =0.0459 \end{aligned} [/latex]

where the subscript [latex]\mathrm{A}[/latex] indicates the data in Table 4.1, and the subscript [latex]\mathrm{B}[/latex] indicates the data in Table 4.8. The comparison of the means for the two analyses is based on the null hypothesis

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

and a two-tailed alternative hypothesis

[latex]H_{\mathrm{A}}: \quad \bar{X}_{\mathrm{A}} eq \bar{X}_{\mathrm{B}}[/latex]

Since the standard deviations can be pooled, the test statistic is calculated using equation 4.20

[latex] t_{ ext {exp }}=\frac{\left|\bar{X}_{\mathrm{A}}-\bar{X}_{\mathrm{B}} ight|}{s_{ ext {pool }} \sqrt{\left(1 / n_{\mathrm{A}}+1 / n_{\mathrm{B}} ight)}}=\frac{|3.117-3.081|}{0.0459 \sqrt{(1 / 7+1 / 5)}}=1.34 [/latex]

The critical value for [latex]t(0.05,10)[/latex], from Appendix [latex]1 \mathrm{~B}[/latex], is 2.23 . Since [latex]t_{\exp }[/latex] is less than [latex]t(0.05,10)[/latex] the null hypothesis is retained, and there is no evidence that the two sets of pennies are significantly different at the chosen significance level.

Penny	Mass (g)
1	3.080
2	3.094
3	3.107
4	3.056
5	3.112
6	3.174
7	3.198

Trial	Volume Delivered (mL)	Trial	Volume Delivered (mL)
1	10.002	6	9.983
2	9.993	7	9.991
3	9.984	8	9.990
4	9.996	9	9.988
5	9.989	10	9.999

Question 4.19: Tables 4.1 and 4.8 show results for two separate experiments...

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:

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

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: