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A normal distribution has a mean of 32 and a standard deviation of 4. Find the probability that a randomly selected x-value from the distribution is in the given interval + 16 16 + 20 24 28 20 24 28 32 X 32 X % % 36 + 40 44 36 40 44

A normal distribution has a mean of 32 and a standard deviation of 4. Find the probability that a randomly selected x-value from the distribution is in the given interval + 16 16 + 20 24 28 20 24 28 32 X 32 X % % 36 + 40 44 36 40 44

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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**Understanding Probabilities in a Normal Distribution** In this educational lesson, we will explore how to find the probability that a randomly selected x-value from a normal distribution falls within a specified interval. Consider the following problem: --- A normal distribution has a mean \( \mu \) of 32 and a standard deviation \( \sigma \) of 4. We want to find the probability that a randomly selected x-value from the distribution falls within a given interval. --- ### Graphical Explanation: 1. **First Interval: 28 to 36** In the first graph, we see a normal distribution curve with a shaded blue area representing the interval from 28 to 36. - The x-axis is labeled with values: 16, 20, 24, 28, 32, 36, 40, 44. - The shaded region lies between 28 and 36. To find the probability that a randomly selected x-value falls within this interval: - Calculate the z-scores for x = 28 and x = 36. - Find the cumulative probabilities for these z-scores using standard normal distribution tables or software. - Subtract the cumulative probability at x = 28 from the cumulative probability at x = 36. Make sure to express the result as a percentage. \[ \text{Probability (P)} = \left[ \text{P}(x \leq 36) - \text{P}(x \leq 28) \right] \times 100 \] Enter the result in the box provided: \[ \_\_\_\_\_\_ \% \] 2. **Second Interval: 24 to 32** In the second graph, we see another normal distribution curve with the shaded blue area representing the interval from 24 to 32. - The x-axis is labeled with values: 16, 20, 24, 28, 32, 36, 40, 44. - The shaded region lies between 24 and 32. To find the probability that a randomly selected x-value falls within this interval: - Calculate the z-scores for x = 24 and x = 32. - Find the cumulative probabilities for these z-scores using standard normal distribution tables or software. - Subtract the cumulative probability at x = 24 from the cumulative probability at x = 32. Make sure to express the result as a percentage
Transcribed Image Text:**Understanding Probabilities in a Normal Distribution** In this educational lesson, we will explore how to find the probability that a randomly selected x-value from a normal distribution falls within a specified interval. Consider the following problem: --- A normal distribution has a mean \( \mu \) of 32 and a standard deviation \( \sigma \) of 4. We want to find the probability that a randomly selected x-value from the distribution falls within a given interval. --- ### Graphical Explanation: 1. **First Interval: 28 to 36** In the first graph, we see a normal distribution curve with a shaded blue area representing the interval from 28 to 36. - The x-axis is labeled with values: 16, 20, 24, 28, 32, 36, 40, 44. - The shaded region lies between 28 and 36. To find the probability that a randomly selected x-value falls within this interval: - Calculate the z-scores for x = 28 and x = 36. - Find the cumulative probabilities for these z-scores using standard normal distribution tables or software. - Subtract the cumulative probability at x = 28 from the cumulative probability at x = 36. Make sure to express the result as a percentage. \[ \text{Probability (P)} = \left[ \text{P}(x \leq 36) - \text{P}(x \leq 28) \right] \times 100 \] Enter the result in the box provided: \[ \_\_\_\_\_\_ \% \] 2. **Second Interval: 24 to 32** In the second graph, we see another normal distribution curve with the shaded blue area representing the interval from 24 to 32. - The x-axis is labeled with values: 16, 20, 24, 28, 32, 36, 40, 44. - The shaded region lies between 24 and 32. To find the probability that a randomly selected x-value falls within this interval: - Calculate the z-scores for x = 24 and x = 32. - Find the cumulative probabilities for these z-scores using standard normal distribution tables or software. - Subtract the cumulative probability at x = 24 from the cumulative probability at x = 32. Make sure to express the result as a percentage
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