How do you use standard deviation for percentages

Question

Example. Potato chip bags normally have a mean of 71 chips with a standard deviation of 2. Approximately what percent of bags contain between 69 and 73 chips?

Answer 1

Let x be the number of chips in a bag. x is normally distributed with mean 71 and standard deviation 2.

We want to find the probability 69 < x < 73 or P(69 < x < 73)

The standardized variable z = (x - mean)/standard deviation = (x - 71/2) is normally distributed with mean 0 and standard deviation 1.

Convert the values of x = 69 and 73 in P(69 < x < 73) to z:

P( (69 - 71)/2 < z < (73-71)/2) = P(-1 < z < 1)

That probability can be found using the standard normal distribution table:

P(-1 < z < 1) = P(z < 1) - P(z < -1) = 0.8413 - 0.1587 = 0.6826