Problem

A machine makes bolts such that the length, $L$ cm, of a bolt has distribution

$$L\sim N(4.1,0.125^2)$$

A bolt is selected at random.

(a) Find the probability that the length of this bolt is more than 4.3 cm.

(b) Show that $P(3.9<L<4.3)$ is 0.890 correct to 3 decimal places.

The machine makes 500 bolts.

The cost to make each bolt is 5 pence.

Only bolts with length between 3.9 cm and 4.3 cm can be used.

These are sold for 9 pence each.

All the bolts that cannot be used are recycled with a scrap value of 1 pence each.

(c) Calculate an estimate for the profit made on these 500 bolts.

Following adjustments to the machine, the length of a bolt, $B$ cm, made by the machine is such that $B\sim N(\mu,\sigma^2)$

Given that $P(B>4.198)=0.025$ and $P(B<4.065)=0.242$

(d) find the value of $\mu$ and the value of $\sigma$

(e) State, giving a reason, whether the adjustments to the machine will result in a decrease or an increase in the profit made on 500 bolts.

中文翻译

某机器生产螺栓，螺栓长度 $L$ cm 服从 $L\sim N(4.1,0.125^2)$。随机选取一枚螺栓。

(a) 求该螺栓长度超过 4.3 cm 的概率。

(b) 证明 $P(3.9<L<4.3)$ 为 0.890（精确至 3 位小数）。

机器生产 500 枚螺栓。每枚制造成本为 5 便士；只有长度介于 3.9 cm 与 4.3 cm 之间的螺栓可用，并以每枚 9 便士出售；不可用的螺栓回收后，每枚废料价值为 1 便士。

(c) 估计这 500 枚螺栓带来的利润。

调整机器后，螺栓长度 $B$ cm 服从 $B\sim N(\mu,\sigma^2)$。已知 $P(B>4.198)=0.025$ 且 $P(B<4.065)=0.242$。

(d) 求 $\mu$ 与 $\sigma$。

(e) 说明机器调整后，生产 500 枚螺栓的利润会减少还是增加，并给出理由。