To find the percentage of observations that lie between [tex]\( z = 0.37 \)[/tex] and [tex]\( z = 1.65 \)[/tex] in a standard normal distribution, we follow these steps:
1. Find the Cumulative Distribution Function (CDF) value for [tex]\( z = 0.37 \)[/tex]:
- The CDF value for [tex]\( z = 0.37 \)[/tex] is [tex]\( 0.6443 \)[/tex]. This means that approximately 64.43% of the observations in a standard normal distribution are less than [tex]\( z = 0.37 \)[/tex].
2. Find the Cumulative Distribution Function (CDF) value for [tex]\( z = 1.65 \)[/tex]:
- The CDF value for [tex]\( z = 1.65 \)[/tex] is [tex]\( 0.9505 \)[/tex]. This means that approximately 95.05% of the observations in a standard normal distribution are less than [tex]\( z = 1.65 \)[/tex].
3. Calculate the percentage of observations between [tex]\( z = 0.37 \)[/tex] and [tex]\( z = 1.65 \)[/tex]:
- To find the percentage of observations that lie between [tex]\( z = 0.37 \)[/tex] and [tex]\( z = 1.65 \)[/tex], we subtract the CDF value for [tex]\( z = 0.37 \)[/tex] from the CDF value for [tex]\( z = 1.65 \)[/tex]:
[tex]\[
0.9505 - 0.6443 = 0.3062
\][/tex]
4. Convert the result to a percentage:
- To express the result as a percentage, we multiply by 100:
[tex]\[
0.3062 \times 100 = 30.62\%
\][/tex]
Thus, approximately 30.62% of the observations lie between [tex]\( z = 0.37 \)[/tex] and [tex]\( z = 1.65 \)[/tex].
Among the provided options, the correct answer is [tex]\( \boxed{30.62\%} \)[/tex].
