An engineer at a bottling facility wants to determine the probability that a randomly chosen bottle has a fill weight that is less than 11.5 ounces, greater than 12.5 ounces, or between 11.5 and 12.5 ounces. The engineer assumes that the bottle fill weights follow a normal distribution with a mean of 12 ounces and a standard deviation of 0.25 ounces.

This example uses the normal distribution. However, you follow the same steps for any distribution that you select.

- In the column name cell of an empty worksheet column, type
`Weight`. - In separate rows, type
`11.5`and`12.5`. These values are the fill weights for which the probabilities will be calculated. - Choose .
- Select Cumulative probability.
- In Mean, enter
`12`. - In Standard deviation, enter
`0.25`. - In Input column, enter
`Weight`. - Click OK.

If the population of fill weights follows a normal distribution and has a mean of 12 and a standard deviation of 0.25, then the following are true:
### Cumulative Distribution Function

- The probability that a randomly chosen bottle has a fill weight that is less than or equal to 11.5 ounces is the CDF at 11.5, which is approximately 0.023.
- The probability that a randomly chosen bottle has a fill weight that is greater than 12.5 ounces is 1 minus the CDF at 12.5, or 1 – 0.977250 = 0.02275.
- The probability that a randomly chosen bottle has a fill weight that is between 11.5 ounces and 12.5 ounces is the CDF at 12.5 minus the CDF at 11.5, or 0.977250 – 0.022750 = 0.954500.

Normal with mean = 12 and standard deviation = 0.25
x P( X ≤ x )
11.5 0.022750
12.5 0.977250