Final answer:
Approximately 92.70% of the values of a normally distributed variable with a mean of 16 and standard deviation of 2 lie between 13 and 21. This is calculated using z-scores and looking up the corresponding percentages on a standard normal distribution table.
Step-by-step explanation:
To find the percentage of all possible values of the normally distributed variable that lies between 13 and 21, we can utilize the properties of the standard normal distribution. First, we need to calculate the z-scores for the given values.
The formula for calculating a z-score is:
Z = (X - μ) / σ
Where:
- Z is the z-score,
- X is the value in the distribution,
- μ is the mean of the distribution,
- σ is the standard deviation of the distribution.
For X = 13:
Z = (13 - 16) / 2 = -1.5
For X = 21:
Z = (21 - 16) / 2 = 2.5
We then look up these z-scores in a standard normal distribution table or use a calculator to find the corresponding percentages. The area under the normal curve between these two z-scores represents the percentage of values between 13 and 21.
With a z-score of -1.5, the percentage is approximately 6.68%, and for a z-score of 2.5, it is approximately 99.38%. To find the percentage between these z-scores, we subtract 6.68% from 99.38%, which gives us:
99.38% - 6.68% = 92.70%
Approximately 92.70% of the values lie between 13 and 21.