To calculate the percentage of values that lie above 2 in a normal distribution with a mean (μ) of 17 and a standard deviation (σ) of 5, we will first determine the z-score for a value of 2. The z-score formula is:
[tex]\[ z = \frac{(X - \mu)}{\sigma} \][/tex]
where:
- [tex]\( X \)[/tex] is the value of interest (in this case, 2),
- [tex]\( \mu \)[/tex] is the mean of the distribution (17),
- [tex]\( \sigma \)[/tex] is the standard deviation (5).
Let's calculate the z-score for [tex]\( X = 2 \)[/tex]:
[tex]\[ z = \frac{(2 - 17)}{5} = \frac{-15}{5} = -3 \][/tex]
A z-score of -3 indicates that the value 2 is 3 standard deviations below the mean.
In a normal distribution, about 99.7% of the data falls within 3 standard deviations (±3σ) from the mean. However, this is the percentage for the values that lie between [tex]\( \mu - 3\sigma \)[/tex] and [tex]\( \mu + 3\sigma \)[/tex] (from about 17 - 15 to 17 + 15).
Since 2 is three standard deviations below the mean, it corresponds to the lower tail end of the distribution. To find the percentage of values that lie above 2, we need to consider the percentage of values from 2 to positive infinity.
Under the empirical rule (also known as the 68-95-99.7 rule), approximately 99.7% of data falls within 3 standard deviations of the mean, which means only about 0.3% of data lies beyond 3 standard deviations from the mean. Since the normal distribution is symmetric, this 0.3% is split equally between the left and right tails of the distribution.
So, if 0.3% of the data lies beyond 3 standard deviations from the mean in both directions, then 0.15% lies beyond 3 standard deviations from the mean in one direction (the one tail).
Therefore, the percentage of values that lie above 2, which is three standard deviations below the mean, should be 100% - 0.15%, which is 99.85%.
As such, approximately 99.85% of values lie above 2 in our normal distribution with a mean of 17 and a standard deviation of 5.
