Answer to Question #305750 in Statistics and Probability for gurpinder

Question

4. The factory makes rulers (like the one in the picture above). Because they use older machinery, there is variability in marking on the ruler – the mean length (distance between 0 and 30 marks) is 30 cm, but it’s normally distributed with the standard deviation of 3 mm.

a. What proportion of the rulers are longer than 30 cm? Show your work. (1)

b. What proportion of the rulers are longer than 30.3 cm? Show your work. (1)

c. Any ruler longer than 30.5 cm or shorter than 29.5 cm are thrown away. What proportion of the rulers is thrown away? Show your work. (1)

d. If 40,000 rulers were made, how many would be thrown away? Show your work. (1)

Answer 1

Let $X=$ the length of the ruler: $X\sim N(30, 0.3^2)$

a.

$$P(X>30)=P(Z>\dfrac{30-30}{0.3})$$ $$=P(Z>0)=0.5$$

$50\%$

b.

$$P(X>30.3)=P(Z>\dfrac{30.3-30}{0.3})$$

$$=P(Z>1)=0.158655$$

$15.87\%$

c.

$$P(29.5<X<30.5)=P(Z<\dfrac{30.5-30}{0.3})$$

$$-P(Z\le\dfrac{29.5-30}{0.3})=P(Z<1.6667)$$

$$-P(Z\le-1.6667)=0.95221-0.04780$$

$$=0.9044$$

$$1-0.9044=0.0956$$

$9.56\%$

d.

$$0.0956(40000)=3824$$

3824 rulers would be thrown away.