Agroup of students weighs 500 us pennies. they find that the pennies have normally distributed weights with a mean of 3.1g and a standard deviation of 0.14g

what percentage of pennies will weigh between 2.8 and 3.3g?
what percentage of pennies will weigh between 2.11 and 3.5g?

what percentage of pennies will weigh less than 2.96g?
what percentage of pennies will weigh more than 3.4g?

Question 1

probability between 2.8 and 3.3

The graph of the normal distribution is shown in the diagram below. We first need to standardise the value of X=2.8 and value X=3.3. Standardising X is just another word for finding z-score

z-score for X = 2.8
(the negative answer shows the position of X = 2.8 on the left of mean which has z-score of 0)

z-score for X = 3.3 


The probability of the value between z=-0.73 and z=0.49 is given by 
P(Z<0.49) - P(Z<-0.73)

P(Z<0.49) = 0.9879
P(Z< -0.73) = 0.2327 (if you only have z-table that read to the left of positive value z, read the value of Z<0.73 then subtract answer from one)

A screenshot of z-table that allows reading of negative value is shown on the second diagram

P(Z<0.49) - P(Z<-0.73) = 0.9879 - 0.2327 = 0.7552 = 75.52%

Question 2
Probability between X=2.11 and X=3.5

z-score for X=2.11


z-score for X=3.5


the probability of P(Z<-2.41) < z < P(Z<0.98) is given by
P(Z<0.98) - P(Z<-2.41) = 0.8365 - 0.0080 = 0.8285 = 82.85%

Question 3
Probability less than X=2.96

z-score of X=2.96

P(Z<-0.34) = 0.3669 = 36.69%

Question 4
Probability more than X=3.4

P(Z>0.73) = 1 - P(Z<0.73) = 1-0.7673=0.2327 = 23.27%