In a certain distribution of numbers, the mean is 100 with a standard deviation of 7. At least what fraction of the numbers are between 37 and 163

To find the fraction of numbers between 37 and 163, we need to first find the z-scores for these values and then use the z-table to find the corresponding probabilities.

The z-score formula is given by:
z = \frac{x - \mu}{\sigma}
where:
- x is the value
- \mu = 100 is the mean
- \sigma = 7 is the standard deviation

For x = 37 :
z_{37} = \frac{37 - 100}{7} = -9

For x = 163 :
z_{163} = \frac{163 - 100}{7} = 9

Next, we look up the z-scores -9 and 9 in the z-table. We find that for a z-score of -9, the probability is close to 0, and for a z-score of 9, the probability is close to 1.

Therefore, the fraction of numbers between 37 and 163 is close to 1 (or 100%).

\textbf{Answer:} \frac{100}{100} = 1