Answered

Discover the answers you need at Westonci.ca, a dynamic Q&A platform where knowledge is shared freely by a community of experts. Discover comprehensive answers to your questions from knowledgeable professionals on our user-friendly platform. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.

In a standard Normal distribution, what percentage of observations lie between [tex][tex]$z=0.37$[/tex][/tex] and [tex][tex]$z=1.65$[/tex][/tex]?

A. [tex]$30.62\%$[/tex]
B. [tex]$40.52\%$[/tex]
C. [tex]$59.48\%$[/tex]
D. [tex]$69.38\%$[/tex]

Sagot :

GinnyAnswer
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]
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.