Steel rods are manufactured with a mean length of 29 centimeter (cm) Because of

Question

Steel rods are manufactured with a mean length of 29 centimeter (cm) Because of variability in the manufacturing process, the lengths of the rods are approximately normally distributed with a standard deviation of 0.06 cm. (a) What proportion of rods has a length less than 28.9 am? (Round to four decimal places as needed.) (b) Any rods that are shorter than 28 87 am or longer than 29.13 cm are discarded What proportion of rods will be discarded? (Round to four decimal places as needed.) (c) Using the results of part (b),if 5000 rods are manufactured in a day, how many should the plant manager expect to discard? (Use the answer from part b to find this answer. Round to the nearest integer as needed.) (d) If an order comes in for 10.000 steel rods, how many rods should the plant manager expect to manufacture if the order states that all rods must be between 28.9 am and 29.1 cm? (Round up to the nearest integer)

Explanation / Answer

Here mean=29 and sd=0.06

a. P(x<28.9)=P(z<28.9-29/0.06)=P(z<-1.67)= 0.0475

b. P(x<28.87 or 29.13<x)=P(-2.167<z or z<2.167)= 0.0302

c. Expected value=0.0302*5000=151

d. from part "a" we got that 4.75% are less than 28.9 cm. Since this is normally distributed, that also means that 4.75 % are more than 29.1 cm (since both measurements are 0.1 cm either side of the mean). Thus 2*4.75% fall outside of the 28.9 cm to 29.1 cm range, or 9.5%.

Let n = number actually manufactured

n(1-discarded percentage) = 10,000
n(1-9.5%) = 10,000
n(0.905) = 10,000
n = 11,049.72, or about 11,050

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