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# Statistics Problems: Normal Distribution, Central Limit Theorem

See attached file.

#1. Z is N 17. I distributed. Find (a) P( Z < .6); (b) P( Z <= -.03);
(c) P( .12 < Z <= 2.14); (d) P( -1.67 <= Z < 1.20)

#2. X is N -10.16 distributed. Find (a) P(X> -12); (b) P(-ll < X <=
-8); (c) P(IX+ 101<2); (d) P(IX+lll> 1.5)

#4. Suppose that the length X of a random bolt produced by machine A is
1.2 em, on the average, with a standard deviation of .3 em. Assuming that
the length X is normally distributed, Find (a) P (X < 1.4); (b) P( X > 1.3);
(c) P( I X - 1.21 >.2)

#6. X is the actual weight of a bag of concrete (labeled "80 pound bag").
Suppose that E(X) = 81 and 6x. =2. Let Y be the average weigth of 10
bags bought: Y=(X + --------- + X ) / 10. (a) What is the distribution ofY
assuming that each bag's weight is normally distributed? Find (b) P (Y =>
80); (c) P( IY - 81.5 1<1).

#### Solution Preview

#1. Given that Z follows Normal distribution with mean zero and standard deviation 1. Symbolically
a) P(Z < 0.60) = 0.7257
b) P( Z ≤ - 0.03) = 0.4880
c) P( 0.12 < Z ≤ 2.14) = P( Z < 2.14) - P( Z < 0.12) = 0.9838 - 0.5478 = 0.4361
d) P ( -1.67 ≤ Z < 1.20) = P( Z < 1.20 ) - P( Z < -1.67) = 0.8849 - 0. 0475 = 0.8375

Note: Note that for continuous distributions P ( Z < k ) = P( Z ≤ ...

#### Solution Summary

Normal distribution and central limit theorem is examined

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