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