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The The width of the slot of a duralumin is (in inches) normally distributed with μ=0.9 and σ=0.003. The specification limits were given as 0.9000±0.0050.
What % of forgings will be defective?

• 0.95
•   0.99
•   0.98
•  0.905

Is it a)  or d)?? Just let me know @Akriti sood..

it is A...can you pls explain this??

for normal distribution,

we have to calculate z-score i.e

and then accordig to this question,P (z1 < X <  z2) which is P  [ ( x1 - u) / σ < X < (x2 - u) /σ ]

this is the formula..right??so what is x1 and x2 here?

if x1 = -0.005 and x2 =0.005 then P [ -0.005 - 0.9 / 0.003  < X <  0.005 - 0.9 / 0.003]..

dun know how to solve further..

See , unless and until we are not told it follows standard normal distribution which relies on finding Z score , we should not do so..We should follow properties of general normal distribution..

According to the question we are considering data about mean = 0.900

And value of one standard deviation                =      ∓ 0.003

But the error mentioned is                              =      ∓ 0.005

which is hence within 2 standard deviations of given data distribution..

We know ,

P(μ - σ  <=   x    <=    μ + σ)  =   0.68

P(μ - 2σ  <=   x    <=    μ + 2σ)  =   0.95

P(μ - 3σ  <=   x    <=    μ + 3σ)  =   1   , where

μ  : Mean of normal distribution

σ : Standard deviation of normal distribution

For standard normal distribution ,

μ  :  0   and  σ  : 1  and   z  =  (x -  μ) /  σ

So as here it is normal distribution we need not find z score..As we can see ,

It follows the 2nd case i.e. under 2 standard deviations but not 1 standard deviation..

Hence the probabiltiy is approximately 0.95 as error limit given is 0.005 but 2 σ = 0.006..

Hence A) is the correct answer..

thanks for the soltuion and correcting me that i should not always follow standard normal distribution.

please explain how did you know that it is within 2 standard deviations from mean?
@habibkhan,if in the question,we were asked to follow standard normal distribution then how would have we solved??
Actually the probability integral is difficult to integrate..
Questions come on general properties here only..
And if some integral comes it will be a trivial one like integrating probability density function from -infinity to +infinity will give 1 as answer..
Things like that will be asked..
@Habib. Its given that if the defect is greater than 0.005 only then it will be considered defective, right?

But 95% which approximates to 0.006 fall within and hence, they are non-defective.

So, only 5% should be defective, right?

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