- Given a normal distribution with a μ = 40 and o u 4.6 find the z-score that corresponds to each x value below. Round all answers to two decimal places. a.) Find the z-score corresponding to x = 37. b.) Find the z-score corresponding to x = 42.
- Given a normal distribution with a μ = 40 and o u 4.6 find the z-score that corresponds to each x value below. Round all answers to two decimal places. a.) Find the z-score corresponding to x = 37. b.) Find the z-score corresponding to x = 42.
MATLAB: An Introduction with Applications
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Author:Amos Gilat
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![### Understanding Z-Scores in a Normal Distribution
A **normal distribution** is a continuous probability distribution for a random variable, where the data tends to cluster around a mean (average). The shape of the distribution is symmetric and bell-shaped.
When dealing with normal distributions, a **z-score** helps us understand how far a particular value (or "x value") is from the mean of the distribution. The formula for calculating the z-score is:
\[ z = \frac{{x - \mu}}{{\sigma}} \]
where:
- \( x \) is the value in question
- \( \mu \) is the mean of the distribution
- \( \sigma \) is the standard deviation of the distribution
Given a normal distribution with a mean (\( \mu \)) of 40 and a standard deviation (\( \sigma \)) of 4.6, find the z-score for each x value provided. Round all answers to two decimal places.
**Problem Set:**
**a.) Find the z-score corresponding to \( x = 37 \).**
\[ \boxed{} \]
**b.) Find the z-score corresponding to \( x = 42 \).**
\[ \boxed{} \]
In these problems:
- You need to substitute the given values into the z-score formula.
- For example, for part (a), substitute \( x = 37 \), \( \mu = 40 \), and \( \sigma = 4.6 \) to find the z-score.
- Compute the value and round to two decimal places.
Ensure you perform the calculations carefully and round off your final answers properly as instructed.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffa81afcb-79d6-433d-9647-f14beb19ad20%2F47ab2625-102b-4580-a3c3-97c2e3372659%2Fqtxsobd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Z-Scores in a Normal Distribution
A **normal distribution** is a continuous probability distribution for a random variable, where the data tends to cluster around a mean (average). The shape of the distribution is symmetric and bell-shaped.
When dealing with normal distributions, a **z-score** helps us understand how far a particular value (or "x value") is from the mean of the distribution. The formula for calculating the z-score is:
\[ z = \frac{{x - \mu}}{{\sigma}} \]
where:
- \( x \) is the value in question
- \( \mu \) is the mean of the distribution
- \( \sigma \) is the standard deviation of the distribution
Given a normal distribution with a mean (\( \mu \)) of 40 and a standard deviation (\( \sigma \)) of 4.6, find the z-score for each x value provided. Round all answers to two decimal places.
**Problem Set:**
**a.) Find the z-score corresponding to \( x = 37 \).**
\[ \boxed{} \]
**b.) Find the z-score corresponding to \( x = 42 \).**
\[ \boxed{} \]
In these problems:
- You need to substitute the given values into the z-score formula.
- For example, for part (a), substitute \( x = 37 \), \( \mu = 40 \), and \( \sigma = 4.6 \) to find the z-score.
- Compute the value and round to two decimal places.
Ensure you perform the calculations carefully and round off your final answers properly as instructed.
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