Question

The refills for a particular mechanical pencil are supposed to be 0.5 mm in diameter. A firm makes refills with diameters whose differences from the correct size are normally distributed with mean 0.5 mm and standard deviation 0.01

a. Suppose you took a sample of 50 pencils. According to the Central Limit Theorem, how would the sample mean of refill thickness vary from sample to sample?

b. Refills below 0.485 mm do not stay in the pencil, while refills above 0.520 mm do not fit in the pencil at all. Determine the probability that a randomly chosen pencil will fit.

Please explain clearly and show ALL working. Thanks!!

Answer 1

Solution:

    We are given that : The refills for a particular mechanical pencil are supposed to be 0.5 mm in diameter. A firm makes refills with diameters whose differences from the correct size are normally distributed with mean 0.5 mm and standard deviation 0.01.

That is : Mean = and Standard Deviation =

Part a) A random sample of 50 pencils is taken. We have to find according to the Central Limit Theorem, how would the sample mean of refill thickness vary from sample to sample?

Thus according to the Central Limit Theorem:

For large sample , sampling distribution of Sample mean follows Normal distribution with Mean =

and standard deviation of sample mean is

Thus the sample mean of refill thickness vary by 0.0014 mm from sample to sample.

Part b) We are given that : Refills below 0.485 mm do not stay in the pencil, while refills above 0.520 mm do not fit in the pencil at all.

Thus refill will be fit in the pencil if its diameter is in between 0.485 mm to 0.520 mm.

We have to find : The probability that a randomly chosen pencil will fit.

That is : P( 0.485 < X < 0.520 ) = ............?

Thus find z score for x = 0.485 and for x = 0.520

z score formula is :

and z score for   x = 0.485

Thus we get :

P( 0.485 < X < 0.520) = P( -1.50 < Z < 1.00)

P( 0.485 < X < 0.520) = P( Z < 1.00 ) - P( Z < -1.50)

Thus look in z table for z= 1.0 and 0.00 as well as look for z = -1.5 and 0.00 and find corresponding area.

For z = 1.0 and 0.00 , corresponding area is : 0.8413

That is : P( Z < 1.00) = 0.8413

For z = -1.5 and 0.00, corresponding area is : 0.0668

That is : P( Z < -1.50) = 0.0668

Thus we get :

P( 0.485 < X < 0.520) = P( Z < 1.00 ) - P( Z < -1.50)

P( 0.485 < X < 0.520) = 0.8413 - 0.0668

P( 0.485 < X < 0.520) = 0.7745

Thus the probability that a randomly chosen pencil will fit is 0.7745 .