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A machine that cuts corks for wine bottles operates in such a way that the distribution of the diameter of the corks produced is well approximated by a normal distribution with mean 4 cm and standard deviation 0.2 cm. The specifications call for corks with diameters between 3.8 and 4.2 cm. A cork not meeting the specifications is considered defective. (A cork that is too small leaks and causes the wine to deteriorate; a cork that is too large doesn't fit in the bottle.) What proportion of corks produced by this machine are defective? (Round the answer to four decimal places.)

Respuesta :

JeanaShupp

Answer: 0.3173

Step-by-step explanation:

Given : A machine that cuts corks for wine bottles operates in such a way that the distribution of the diameter of the corks produced is well approximated by a normal distribution with

[tex]\mu=4\ cm[/tex] and [tex]\sigma=0.2\ cm[/tex]

The specifications call for corks with diameters between 3.8 and 4.2 cm.

Let x be the random variable that represents the  the diameter of the corks.

Using formula [tex]z=\dfrac{x-\mu}{\sigma}[/tex], the z-score corresponding to x= 3.8 will be :_

[tex]z=\dfrac{3.8-4}{0.2}=1[/tex]

z-score corresponding to x= 4.2 will be :_

[tex]z=\dfrac{4.2-4}{0.2}=1[/tex]

Now, by using the standard normal distribution table for z, we have

[tex]\text{P value}=P(-1<z<1)=2P(z<1)-1\\\\=2(0.8413447)-1\\\\=0.6826894\approx0.6827[/tex]

∴The proportion of corks produced by this machine are meeting the specifications=0.6827

∴The proportion of corks produced by this machine are defective = [tex]1-0.6827=0.3173[/tex]

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