Question

Thompson and Thompson is a steel bolts manufacturing company. Their current steel bolts have a mean diameter of 145145 millimeters, and a standard deviation of 77 millimeters. If a random sample of 3131 steel bolts is selected, what is the probability that the sample mean would differ from the population mean by greater than 33 millimeters

Answer 1

Answer:

The probability that the sample mean would differ from the population mean by greater than 33 millimeters is 0.0174

Step-by-step explanation:

$Mean = \mu = 145 mm$

Standard deviation =$\sigma = 7$

We are supposed to find the probability that the sample mean would differ from the population mean by greater than 3 millimeters

$P(|x-\mu|>33)=1-P(|x-\mu|<3)\\P(|x-\mu|>33)=1-P(-3<|x-\mu|<3)\\P(|x-\mu|>33)=1-P(\frac{-3}{\frac{\sigma}{\sqrt{n}}}<\frac{|x-\mu|}{\frac{\sigma}{\sqrt{n}}}<\frac{-3}{\frac{\sigma}{\sqrt{n}}})\\P(|x-\mu|>33)=1-P(\frac{-3}{\frac{7}{\sqrt{31}}}<\frac{|x-\mu|}{\frac{\sigma}{\sqrt{n}}}<\frac{-3}{\frac{7}{\sqrt{31}}})\\P(|x-\mu|>33) =1-P(-2.38<z<2.38)\\P(|x-\mu|>33) =1-(P(z<2.38)-P(z<-2.38))$

Using Z table

                   = 1-(0.9913-0.0087)

                  =0.0174

Hence  the probability that the sample mean would differ from the population mean by greater than 33 millimeters is 0.0174