Alright, let's go through the process of solving this question step-by-step.
### Understanding the Problem
We are given two z-scores, [tex]\( z = 0.37 \)[/tex] and [tex]\( z = 1.65 \)[/tex], and we need to find the percentage of observations that lie between these two z-scores in a standard normal distribution.
### Standard Normal Distribution
A standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. The z-score represents the number of standard deviations a point is from the mean.
### Z-Scores and Cumulative Distribution Function (CDF)
The cumulative distribution function (CDF) for a z-score gives us the probability that a standard normal random variable will be less than or equal to that z-score.
### Finding Probabilities
1. For [tex]\( z = 0.37 \)[/tex]:
The CDF value (probability) corresponding to [tex]\( z = 0.37 \)[/tex] is approximately 0.6443 (or 64.43%).
2. For [tex]\( z = 1.65 \)[/tex]:
The CDF value (probability) corresponding to [tex]\( z = 1.65 \)[/tex] is approximately 0.9505 (or 95.05%).
### Calculating the Percentage Between the Two Z-Scores
The percentage of observations that lie between [tex]\( z = 0.37 \)[/tex] and [tex]\( z = 1.65 \)[/tex] is found by subtracting the probability at [tex]\( z = 0.37 \)[/tex] from the probability at [tex]\( z = 1.65 \)[/tex] and then converting this probability to a percentage.
1. Subtracting the Probabilities:
[tex]\( 0.9505 - 0.6443 = 0.3062 \)[/tex]
2. Converting to Percentage:
[tex]\( 0.3062 \times 100 = 30.62\% \)[/tex]
### Conclusion
Therefore, the percentage of observations that lie between [tex]\( z = 0.37 \)[/tex] and [tex]\( z = 1.65 \)[/tex] in a standard normal distribution is [tex]\( 30.62\% \)[/tex].
Given the options provided:
[tex]\( 30.62 \% \)[/tex]
[tex]\( 40.52 \% \)[/tex]
[tex]\( 59.48 \% \)[/tex]
[tex]\( 69.38 \% \)[/tex]
The correct answer is:
[tex]\[ \boxed{30.62\%} \][/tex]
