To determine the percentage of observations that lie between [tex]\( z = 0.37 \)[/tex] and [tex]\( z = 1.65 \)[/tex] in a standard normal distribution, we can use the cumulative distribution function (CDF) values for these z-scores.
1. Find the cumulative probabilities:
- For [tex]\( z = 0.37 \)[/tex], the cumulative probability (denoted as [tex]\( P(Z \leq 0.37) \)[/tex]) is approximately 0.6443. This means that about 64.43% of the data lies to the left of [tex]\( z = 0.37 \)[/tex].
- For [tex]\( z = 1.65 \)[/tex], the cumulative probability (denoted as [tex]\( P(Z \leq 1.65) \)[/tex]) is approximately 0.9505. This means that about 95.05% of the data lies to the left of [tex]\( z = 1.65 \)[/tex].
2. Calculate the percentage of observations between the z-scores:
- To find the percentage of observations between [tex]\( z = 0.37 \)[/tex] and [tex]\( z = 1.65 \)[/tex], we need to subtract the cumulative probability at [tex]\( z = 0.37 \)[/tex] from the cumulative probability at [tex]\( z = 1.65 \)[/tex].
[tex]\[
P(0.37 \leq Z \leq 1.65) = P(Z \leq 1.65) - P(Z \leq 0.37)
\][/tex]
[tex]\[
P(0.37 \leq Z \leq 1.65) = 0.9505 - 0.6443 = 0.3062
\][/tex]
3. Convert the decimal to a percentage:
- Convert the decimal [tex]\( 0.3062 \)[/tex] to a percentage by multiplying by 100:
[tex]\[
0.3062 \times 100 = 30.62\%
\][/tex]
Therefore, the percentage of observations lying between [tex]\( z=0.37 \)[/tex] and [tex]\( z=1.65 \)[/tex] is [tex]\( 30.62\% \)[/tex].
Among the given options:
[tex]$30.62 \%$[/tex]
[tex]$40.52 \%$[/tex]
[tex]$59.48 \%$[/tex]
[tex]$69.38 \%$[/tex]
The correct answer is:
[tex]$30.62 \%$[/tex]
