Question

Due to random variations in the operation of an automatic coffee​ machine, not every cup is filled with the same amount of coffee. Assume that the mean amount of coffee dispensed is 8 ounces and the standard is deviation is 0.5 ounce. Use the 65-95-99.7 rule to complete the following.

a. Approximately ___ % of cups should have less than 8 ounces?

b. Approximately ___% of cups should have less than 6.5 ounces?

Answer 1

It is not 65-95-99.7 rule but it is 68-95-99.7 rule

68-95-99.7 rule states that

There will be 68% of values within one standard deviation from the mean on either side i.e between - and + there will be 68% of values

There will be 95% of values within two standard deviation from the mean on either side i.e between - 2 and + 2 there will be 95% of values

There will be 99.7% of values within three standard deviation from the mean on either side i.e between - 3 and + 3 there will be 99.7% of values

Given that mean amount of coffee dispensed is 8 ounces and the standard is deviation is 0.5 ounce

= 8, = 0.5

Question (a)

We need to find the percentage of cups that have less than 8 ounces

Since mean is 8 here, there will be 50% values to left of mean and 50% values to the right of mean

So Approximately 50% of cups should have less than 8 ounces

Question (b)

We need to find the percentage of cups that have less than 6.5 ounces

Since mean is 8 here and standard deviation is 0.5,

6.5 = 8 - 3 * 0.5

= - 3 *

So we need to find the values that are left to - 3 *

We know there will be 99.7% values between - 3 * and + 3 *

So there will 0.3% values that are not between   - 3 * and + 3 *

There will be 0.15% values on both sides of the mean since the curve is symmetrical about mean

So Approximately 0.15% of  of cups should have less than 6.5 ounces