Suppose 4

Suppose that 14% of all steel shafts produced by a certain process are nonconforming but can be reworked (rather than having to be scrapped). Consider a random sample of 200 shafts, and let X denote the number among these that are nonconforming and can be reworked.
a) What is the probability that X is at most 30?  
b) What is the probability that X is less than 30?  
c) What is X's probability of being between 15 and 25 (inclusive)?

Correct answer:

a =  0.6948
b =  0.6201
c =  0.3022

Step-by-step explanation:

$$\begin{gathered} n=200 \\ p=14\%=\frac{14}{100}=\frac{7}{50}=0.14 \\ \mu =n\cdot p=200\cdot \frac{7}{50}=\frac{200\cdot 7}{50}=\frac{1400}{50}=28 \\ \sigma =\sqrt{\mu \cdot (1-p)}=\sqrt{28\cdot (1-0.14)}\doteq 4.9071 \\ {z}_1=\frac{30+1/2-\mu }{\sigma }=\frac{30+1/2-28}{\sigma }\doteq 0.5095 \\ a=0.6948 \end{gathered}$$

Use the normal distribution table.

$$\begin{gathered} z_2=\frac{30-1/2-\mu }{\sigma }=\frac{30-1/2-28}{\sigma }\doteq 0.3057 \\ b=0.6201 \end{gathered}$$

$c=0.3022$

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