Respuesta :
Answer:
DIRECT WAY EXCEL
"=NORM.DIST(75,80,3,TRUE)"
And we got: [tex] P(X\leq 75)= 0.04779[/tex]
OTHER WAY
[tex]P(X\leq 75)=P(\frac{X-\mu}{\sigma}\leq \frac{75-\mu}{\sigma})=P(Z\leq \frac{75-80}{3})=P(Z<-1.67)[/tex]
And we can find this probability using the normal standard table or excel:
[tex]P(Z<-1.67)=0.04779[/tex]
Step-by-step explanation:
Previous concepts
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean". Â
Solution to the problem
Let X the random variable that represent the diameter of a population, and for this case we know the distribution for X is given by:
[tex]X \sim N(80,3)[/tex] Â
Where [tex]\mu=80[/tex] and [tex]\sigma=3[/tex]
And we know that if the diameter is 75 or less the ring would be considered defective , so then in order to find the proportion of defective we need to find the following probability:
[tex] P(X\leq 75)[/tex]
One way to do this in excel is with the following formula:
"=NORM.DIST(75,80,3,TRUE)"
And we got: [tex] P(X\leq 75)= 0.04779[/tex]
And the other way is use the z score formula given by:
[tex]z=\frac{x-\mu}{\sigma}[/tex]
If we apply this formula to our probability we got this:
[tex]P(X\leq 75)=P(\frac{X-\mu}{\sigma}\leq \frac{75-\mu}{\sigma})=P(Z\leq \frac{75-80}{3})=P(Z<-1.67)[/tex]
And we can find this probability using the normal standard table or excel:
[tex]P(Z<-1.67)=0.04779[/tex]
