2 The weight X of a metal ingot produced in a casting foundr

2. The weight, X, of a metal ingot produced in a casting foundry follows normal distribution. The mean and standard deviation of X are mu = 520 pounds and sigma = 11 pounds respectively. Find the following: (10 points) a. P(X

Solution

= p(Z<= 525-520/11)

=p(Z<=0.45)

From the Z tables, we see that this probability is 0.5+0.1736= 0.6736

(Ans)

2. P(509<=X<=531) = ?

Converting into Z values, we have, z1= 509-520/11 =-1

Then, z2= 531-520/11= 1

Then, p(-1<Z<1) = 0.3413+0.3413= 0.6826

(ans)

3. The central 98% probability of limits can be found from the Z tables,

Checking for the Z values that correspond to the areas of 0.49 on either side, we get that the z scores are 2.33 and -2.33

Finding the corresponding X values, we have, 2.33= X-520/11

Then, -2.33= X-520/11 = > X = 494.37

Thus, the 98% probability limits are 494.37 and 545.63 (ans)