Rockwell hardness of pins of a certain type is known to have a mean value of 50 and a standard deviation of 1.2.a. If the distribution is normal, what is the probability that the sample mean hardness for a random sample of 9 pins is at least 51?b. Without assuming population normality, what is the (approximate) probability that the sample mean hardness for a random sample of 40 pins is at least 51?

a

b

Step-by-step explanation:

From the question we are told that

The mean value is

The standard deviation is  

Considering question a

The sample size is  n = 9

Generally the standard error of the mean is mathematically represented as

=>  

=>  

Generally the probability that the sample mean hardness for a random sample of 9 pins is at least 51 is mathematically represented as

=>  

From the z table  the area under the normal curve to the left corresponding to  2.5  is

=>

=>

Considering question b

The sample size is  n = 40

   Generally the standard error of the mean is mathematically represented as

=>  

=>  

Generally the (approximate) probability that the sample mean hardness for a random sample of 40 pins is at least 51 is mathematically represented as  

=>

=>  

From the z table  the area under the normal curve to the left corresponding to  5.2715 and

=>  

So

=>

circle area = pi * radius^2

if diameter = 10 then radius = 5

circle area = pi * 5^2

circle area = pi * 25

circle area = 78.5398163397 square units

step-by-step explanation:

it is true

step-by-step explanation:

all the combinations add up i used   calculator