Question 24.14: Tension test is carried out on 120 specimens made of grey ca...

\text { Given } \mu=300 N / mm ^{2} \quad \hat{\sigma}=25 N / mm ^{2} .

Step I Number of specimens having UTS less than 275 N/mm².

X_{1}=275 N / mm ^{2} .

Z_{1}=\frac{X_{1}-\mu}{\hat{\sigma}}=\frac{275-300}{25}=-1 .

As shown in Fig. 24.20(a), the area below the normal curve from Z_{1}=-1 \text { to } Z=-\infty indicates the probability of specimens having UTS less than 275 N/mm². The normal curve is symmetrical about Y-axis. Therefore, the area below the normal curve from Z = 0 to Z = –1 is equal to the area below the curve from Z = 0 to Z = + 1. From Table 24.6, the area below normal curve from Z = 0 to Z = 1 is 0.3413. Also, the area below normal curve from Z = –∞ to Z = 0 is 0.5. Therefore, Shaded area in Fig. 24.20(a) = 0.5 – 0.3413
= 0.1587.

Table 24.6 Areas under normal curve from 0 to Z

Therefore, 15.87% of specimens will have UTS less than 275 N/mm².

No. of specimens = 0.1587 × 120 = 19.04 or 19               (i).

Step II Number of specimens having UTS between 275 and 350 N/mm²

X_{2}=350 N / mm ^{2} .

Z_{2}=\frac{X_{2}-\mu}{\hat{\sigma}}=\frac{350-300}{25}=+2 .

As shown in Fig. 24.20(b), the area below the normal curve from Z_{1}=-1 \text { to } Z_{2}=+2 indicates the probability of specimens having UTS between 275 and 350 N/mm². From Table 24.6, the area below normal curve from Z = 0 to Z = +2 is 0.4772.
Shaded area in Fig. 24.20(b) = 0.3413 + 0.4772 = 0.8185.
Therefore, 81.85% of specimens will have UTS between 275 to 350 N/mm².
No. of specimens = 0.8185 × 120 = 98.22 or 98                   (ii)