The amount of Jen's monthly phone bill is normally distributed with a mean of $50 and a standard deviation of $12. What percentage of her phone bills are between $14 and $86?
the z score for $14 is (14-50)/12 = -3
the z score for $86 = (86-50)/12 = +3
at this point you will need a table of values of z scores, or some distribution calculator
Here is a rather simple one
http://davidmlane.com/hyperstat/z_table.html
You could either enter
mean: 0
standard deviation: 1
between -3 and +3
or for this one we didn't even have to calculate the z-scores and could have entered
Mean 50
SD 12
between 14 and 86, notice we got the same result, .9973 which is 99.73%
To find the percentage of Jen's phone bills that are between $14 and $86, we need to calculate the area under the normal distribution curve between these two values.
Step 1: Standardize the values
To standardize the values, we subtract the mean and then divide by the standard deviation.
Standardized value for $14: Z1 = (14 - 50) / 12 = -36 / 12 = -3
Standardized value for $86: Z2 = (86 - 50) / 12 = 36 / 12 = 3
Step 2: Find the corresponding cumulative probabilities
Now we need to find the cumulative probabilities associated with these standardized values. We can use a standard normal distribution table or a calculator to find the probabilities.
Using a standard normal distribution table or calculator, the cumulative probability for Z = -3 is approximately 0.0013.
The cumulative probability for Z = 3 is also approximately 0.9987.
Step 3: Calculate the percentage
To find the percentage between $14 and $86, we subtract the cumulative probability for Z = -3 from the cumulative probability for Z = 3 and multiply by 100.
Percentage = (0.9987 - 0.0013) * 100 = 0.9974 * 100 = 99.74%
Therefore, approximately 99.74% of Jen's phone bills are between $14 and $86.