To find the fraction of numbers that are less than 35 or more than 85 in a distribution with a mean of 60 and a standard deviation of 5, follow these steps:
1. Calculate the z-score for the lower bound (35):
The z-score measures how many standard deviations an element is from the mean. The formula for the z-score is:
[tex]\[
z = \frac{X - \mu}{\sigma}
\][/tex]
where [tex]\(X\)[/tex] is the value, [tex]\(\mu\)[/tex] is the mean, and [tex]\(\sigma\)[/tex] is the standard deviation.
For the lower bound (35):
[tex]\[
z_{\text{lower}} = \frac{35 - 60}{5} = \frac{-25}{5} = -5.0
\][/tex]
2. Calculate the z-score for the upper bound (85):
Using the same formula:
[tex]\[
z_{\text{upper}} = \frac{85 - 60}{5} = \frac{25}{5} = 5.0
\][/tex]
3. Find the cumulative probability for each z-score:
The cumulative distribution function (CDF) for a standard normal distribution provides the probability that a standard normal random variable will be less than or equal to a given value.
For [tex]\( z_{\text{lower}} = -5.0 \)[/tex]:
[tex]\[
\text{prob}_{\text{lower}} \approx 2.866515718791933 \times 10^{-7}
\][/tex]
Since [tex]\( z_{\text{upper}} = 5.0 \)[/tex] is symmetric about the mean:
[tex]\[
\text{prob}_{\text{upper}} = 1 - \text{CDF}(z_{\text{upper}}) \approx 2.866515719235352 \times 10^{-7}
\][/tex]
4. Calculate the total fraction of numbers less than 35 or more than 85:
Add the probabilities from the lower and upper bounds:
[tex]\[
\text{fraction} = \text{prob}_{\text{lower}} + \text{prob}_{\text{upper}} \approx 2.866515718791933 \times 10^{-7} + 2.866515719235352 \times 10^{-7} \approx 5.733031438027284 \times 10^{-7}
\][/tex]
Thus, the largest fraction of the numbers that are either less than 35 or more than 85 is:
[tex]\[ \boxed{5.733031438027284 \times 10^{-7}} \][/tex]
