The tolerance specified by the designer for the diameters of a transmission shaft is 25.000 ± 0.025 mm. The shafts are machined on three different machines. It was observed from the sample of shafts that the diameters are normally distributed with a standard deviation of 0.015 mm for each of the

Question

The tolerance specified by the designer for the diameters of a transmission shaft is 25.000 ± 0.025 mm. The shafts are machined on three different machines. It was observed from the sample of shafts that the diameters are normally distributed with a standard deviation of 0.015 mm for each of the three machines. However, the mean diameter of shafts fabricated on the three machines is found to be 24.99, 25.00 and 25.01 mm respectively. Determine the percentage of rejected shafts in each case and comment on the result.

Accepted Answer

[latex] ext { Given } \hat{\sigma}=0.015 mm \quad X_{1}=25.000-0.025 mm [/latex].

[latex] X_{2}=25.000+0.025 mm \quad \mu_{1}=24.99 mm [/latex].

[latex] \mu_{2}=25 mm \quad \mu_{3}=25.01 mm [/latex].

Step I Percentage of rejected shafts in each machine
The lower and upper limits specified by the designer are,

[latex] X_{1}=25.000-0.025=24.975 mm [/latex].

[latex] X_{2}=25.000+0.025=25.025 mm [/latex].

The results are tabulated in the following way:

Machine C	Machine B	Machine A
25.01	25.00	24.99	mean μ (mm)
–2.33	-1.67	-1	[latex] Z_{1}=\frac{X_{1}-\mu}{\hat{\sigma}} [/latex]
+1	+1.67	+2.33	[latex]Z_{2}=\frac{X_{2}-\mu}{\hat{\sigma}}[/latex]
0.4901	0.4525	0.3413	[latex] ext { Area } A_{1}[/latex] [latex]\left(Z=0 ext { to } Z_{1} ight)[/latex]
0.3413	0.4525	0.4901	[latex] ext { Area } A_{2}[/latex] [latex]\left(Z=0 ext { to } Z_{2} ight)[/latex]
0.8314	0.9050	0.8314	[latex]\left(A_{1}+A_{2} ight)[/latex]
16.86	9.5	16.86	Percentage rejection

Step II Comments on result
The above data is illustrated in Fig. 24.13. It is observed from the figure that in case of machines A or C, the process is not centred, which results in a large percentage of rejection. With the same magnitude of tolerance, the percentage of rejection is small in case of the machine B, because the process is centred.

Question 24.4: The tolerance specified by the designer for the diameters of...

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