Bass Bearings

please help me with stat hw question?

Ball bearings are manufactured with a mean diameter of 2.20 millimeters (mm). Because of variability in the manufacturing process, the diameters of the bass bearings are approximately normally distributed with a standard deviation of 0.98 mm.
a. What proportion of ball bearings has a diameter more than 3.13 mm?

b. Any ball bearings that have a diameter greater than 2.69 mm or less than 1.66 mm are discarded. What proportion of ball bearings will be discarded? (2 points)

c. If the company needs 200 acceptable ball bearings, how many should be manufactured? (2 points)

.. P(B > 3.13)
= 1 - P(B ≤ 3.13)
= 1 - P(Z ≤ (3.13 - 2.2) / 0.98)
= 1 - phi(0.93/0.98)
= 1 - phi(0.94897...)
= 1 - 0.82868
= 0.17132
0.17132 of ball bearings will have a diameter greater than 3.13

.. P(B < 1.66 or B > 2.69)
= 1 - P(1.66 ≤ B ≤ 2.69)
= 1 - { P(B ≤ 2.69) - P(B ≤ 1.66) }
= 1 - { P(Z ≤ (2.69 - 2.2) / 0.98) - P(Z ≤ (1.66 - 2.2) / 0.98) }
= 1 - { P(Z ≤ 0.5) - P(Z ≤ -0.55102...) }
= 1 - { P(Z ≤ 0.5) - [ 1 - P(Z ≤ 0.55102...) ] }
= 1 - { phi(0.5) - 1 + phi(0.55102...) }
= 1 - phi(0.5) + 1 - phi(0.55102...)
= 2 - phi(0.5) - phi(0.55102...)
= 2 - 0.69146 - 0.70919
= 0.59935
Thus almost 60% of ball bearings produced will be discarded. Wow!!

If only 40% of ball bearings are acceptable, then your target of 200 is that 40%, thus the full 100% must be:
x * 40% = 200
x = 200 / 40%
x = 200 / (40/100)
x = 200*100/40
x = 500
They will need to produce 500 ball bearings to produce the required 200 acceptable ones, assuming the distribution is accurate.

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