Question

Steel rods are manufactured with a mean length of 26 centimeter​ (cm). Because of variability in the manufacturing​ process, the lengths of the rods are approximately normally distributed with a standard deviation of 0.09cm. Complete parts ​(a) to​(d).

​(a) What proportion of rods has a length less than 25.9 ​cm?

​(b) Any rods that are shorter than 25.82 cm or longer than 26.18cm are discarded. What proportion of rods will be​ discarded?

​(c) Using the results of part (b)​if 5000rods are manufactured in a​ day, how many should the plant manager expect to​ discard?

​(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 25.9cm and 26.1​cm?

Answer 1

Given,

We know that,

(a)

Proportion of rods has a length less than 25.9 ​cm is,

z = (25.9 - 26)/0.09

= - 1.1111

P(x < 25.9) = P(z < -1.1111) = 0.1333

= 13.33%

(b)

Rods that are shorter than 25.82 cm or longer than 26.18cm are discarded. The proportion of rods that will be​ discarded is,

z_25.82 = -2

z_26.18 =herefore,

P(25.82 > x > 26.18) = P(-2 < z < 2) = 0.0455

  = 4.55%

(c)

​if 5000 rods are manufactured in a​ day, the plant manager should expect to​ discard,

= 5000 * 0.0455

= 228 rods

(d)

z_25.9 = (25.9 - 26)/0.09 = -1.1111

z_26.1 = (26.1 - 26)/0.09 = 1.1111

P(26.9 < x < 26.1) = P(-1.11 < z < 1.11) = 0.7335 = 73.35%

Therefore, if only 73.35% satisfy the condition of the order, to get the 10,000 rods that satisfy the condition, the total number of rods that should be manufactured is,

= 10000/0.7335

= 13633 rods