The amount of aspirin in the analgesic tablets from a particular manufacturer is known to follow a normal distribution, with [latex]\mu=250 ~\mathrm{mg}[/latex] and [latex]\sigma^{2}=25[/latex]. In a random sampling of tablets from the production line, what percentage are expected to contain between 243 and [latex]262 ~\mathrm{mg}[/latex] of aspirin?

The normal distribution for this example is shown in Figure 4.7, with the shaded area representing the percentage of tablets containing between 243 and [latex]262~ \mathrm{mg}[/latex] of aspirin. To determine the percentage of tablets between these limits, we first determine the percentage of tablets with less than [latex]243~ \mathrm{mg}[/latex] of aspirin, and the percentage of tablets having more than [latex]262 ~\mathrm{mg}[/latex] of aspirin. This is accomplished by calculating the deviation, [latex]z[/latex], of each limit from [latex]\mu[/latex], using the following equation [latex]z=\frac{X-\mu}{\sigma}[/latex] where [latex]X[/latex] is the limit in question, and [latex]\sigma[/latex], the population standard deviation, is 5 . Thus, the deviation for the lower limit is [latex]z_{\text {low }}=\frac{243-250}{5}=-1.4[/latex] and the deviation for the upper limit is [latex]z_{\text {up }}=\frac{262-250}{5}=+2.4[/latex] Using the table in Appendix 1A, we find that the percentage of tablets with less than [latex]243~ \mathrm{mg}[/latex] of aspirin is [latex]8.08 \%[/latex], and the percentage of tablets with more than [latex]262~ \mathrm{mg}[/latex] of aspirin is [latex]0.82 \%[/latex]. The percentage of tablets containing between 243 and [latex]262~ \mathrm{mg}[/latex] of aspirin is therefore [latex]100.00 \%-8.08 \%-0.82 \%=91.10 \%[/latex]

Question 4.11: The amount of aspirin in the analgesic tablets from a partic...

Modern Analytical Chemistry

The amount of aspirin in the analgesic tablets from a particular manufacturer is known to follow a normal distribution, with $\mu=250 ~\mathrm{mg}$ and $\sigma^{2}=25$ . In a random sampling of tablets from the production line, what percentage are expected to contain between 243 and $262 ~\mathrm{mg}$ of aspirin?

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The normal distribution for this example is shown in Figure 4.7, with the shaded area representing the percentage of tablets containing between 243 and $262~ \mathrm{mg}$ of aspirin. To determine the percentage of tablets between these limits, we first determine the percentage of tablets with less than $243~ \mathrm{mg}$ of aspirin, and the percentage of tablets having more than $262 ~\mathrm{mg}$ of aspirin. This is accomplished by calculating the deviation, $z$ , of each limit from $\mu$ , using the following equation

$z=\frac{X-\mu}{\sigma}$

where $X$ is the limit in question, and $\sigma$ , the population standard deviation, is 5 . Thus, the deviation for the lower limit is

$z_{\text {low }}=\frac{243-250}{5}=-1.4$

and the deviation for the upper limit is

$z_{\text {up }}=\frac{262-250}{5}=+2.4$

Using the table in Appendix 1A, we find that the percentage of tablets with less than $243~ \mathrm{mg}$ of aspirin is $8.08 \%$ , and the percentage of tablets with more than $262~ \mathrm{mg}$ of aspirin is $0.82 \%$ . The percentage of tablets containing between 243 and $262~ \mathrm{mg}$ of aspirin is therefore

$100.00 \%-8.08 \%-0.82 \%=91.10 \%$

Appendix 1A
Single-Sided Normal Distribution $^a$
u	0.00	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09
0.0	0.5000	0.4960	0.4920	0.4880	0.4840	0.4801	0.4761	0.4721	0.4681	0.4641
0.1	0.4602	0.4562	0.4522	0.4483	0.4443	0.4404	0.4365	0.4325	0.4286	0.4247
0.2	0.4207	0.4168	0.4129	0.4090	0.4052	0.4013	0.3974	0.3936	0.3897	0.3859
0.3	0.3821	0.3783	0.3745	0.3707	0.3669	0.3632	0.3594	0.3557	0.3520	0.3483
0.4	0.3446	0.3409	0.3372	0.3336	0.3300	0.3264	0.3228	0.3192	0.3156	0.3121
0.5	0.3085	0.3050	0.3015	0.2981	0.2946	0.2912	0.2877	0.2843	0.2810	0.2776
0.6	0.2743	0.2709	0.2676	0.2643	0.2611	0.2578	0.2546	0.2514	0.2483	0.2451
0.7	0.2420	0.2389	0.2358	0.2327	0.2296	0.2266	0.2236	0.2206	0.2177	0.2148
0.8	0.2119	0.2090	0.2061	0.2033	0.2005	0.1977	0.1949	0.1922	0.1894	0.1867
0.9	0.1841	0.1814	0.1788	0.1762	0.1736	0.1711	0.1685	0.1660	0.1635	0.1611
1.0	0.1587	0.1562	0.1539	0.1515	0.1492	0.1469	0.1446	0.1423	0.1401	0.1379
1.1	0.1357	0.1335	0.1314	0.1292	0.1271	0.1251	0.1230	0.1210	0.1190	0.1170
1.2	0.1151	0.1131	0.1112	0.1093	0.1075	0.1056	0.1038	0.1020	0.1003	0.0985
1.3	0.0968	0.0951	0.0934	0.0918	0.0901	0.0885	0.0869	0.0853	0.0838	0.0823
1.4	0.0808	0.0793	0.0778	0.0764	0.0749	0.0735	0.0721	0.0708	0.0694	0.0681
1.5	0.0668	0.0655	0.0643	0.0630	0.0618	0.0606	0.0594	0.0582	0.0571	0.0559
1.6	0.0548	0.0537	0.0526	0.0516	0.0505	0.0495	0.0485	0.0475	0.0465	0.0455
1.7	0.0446	0.0436	0.0427	0.0418	0.0409	0.0401	0.0392	0.0384	0.0375	0.0367
1.8	0.0359	0.0351	0.0344	0.0336	0.0329	0.0322	0.0314	0.0307	0.0301	0.0294
1.9	0.0287	0.0281	0.0274	0.0268	0.0262	0.0256	0.0250	0.0244	0.0239	0.0253
2.0	0.0228	0.0222	0.0217	0.0212	0.0207	0.0202	0.0197	0.0192	0.0188	0.0183
2.1	0.0179	0.0174	0.0170	0.0166	0.0162	0.0158	0.0154	0.0150	0.0146	0.0143
2.2	0.0139	0.0136	0.0132	0.0129	0.0125	0.0122	0.0119	0.0116	0.0113	0.0110
2.3	0.0107	0.0104	0.0102		0.00964		0.00914		0.00866
2.4	0.00820		0.00776		0.00734		0.00695		0.00657
2.5	0.00621		0.00587		0.00554		0.00523		0.00494
2.6	0.00466		0.00440		0.00415		0.00391		0.00368
2.7	0.00347		0.00326		0.00307		0.00289		0.00272
2.8	0.00256		0.00240		0.00226		0.00212		0.00199
2.9	0.00187		0.00175		0.00164		0.00154		0.00144
3.0	0.00135
3.1	0.000968
3.2	0.000687
Single-Sided Normal Distribution $^a$ —continued
u	0.00	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09
3.3	0.000483
3.4	0.000337
3.5	0.000233
3.6	0.000159
3.7	0.000108
3.8	0.0000723
3.9	0.0000481
4.0	0.0000317
4.1	0.0000207
4.2	0.0000133
4.3	0.00000854
4.4	0.00000541
4.5	0.00000340
4.6	0.00000211
4.7	0.00000130
4.8	0.000000793
4.9	0.000000479
5.0	0.000000287

$^a$ This table gives the proportion, P, of the area under a normal distribution curve that lies to the right of the deviation z, where z is defined as

z = (X – µ)/σ

For example, the proportion of the area under a normal distribution curve that lies to the right of a deviation of 0.04 is 0.4840, or 48.40%. The area to the left of the deviation is given as 1 – P. Thus, 51.60% of the area under the normal distribution curve lies to the left of a deviation of 0.04. When the deviation is negative, the values in the table give the proportion of the area under the normal distribution curve that lies to the left of z; therefore, 48.40% of the area lies to the left, and 51.60% of the area lies to the right of a deviation of –0.04.