**Transcription for Educational Content**---**Normal Distribution and Probability Density Functions**In the figure above, we are presented with four graphs depicting different probability density functions modeled by normal curves. The task is to sketch a normal curve and label the horizontal axis with values of 45, 50, 55, 60, 65, 70, and 75.Each graph displays a bell-shaped curve, characteristic of normal distribution, which is symmetrical around the mean. **Graphs Explanation:**1. **Graph 1:** The mean appears to be at 60, with the horizontal axis labeled from 75 to 45.2. **Graph 2:** The mean appears to be at 65, with the horizontal axis labeled from 45 to 75.3. **Graph 3:** The mean seems to be 60, with the axis values ranging from 45 to 75.4. **Graph 4:** The mean looks to be at 65, with labels from 75 to 45.The goal is to identify the correct arrangement of horizontal axis labels as stated in the task: 45, 50, 55, 60, 65, 70, and 75.**Probability Questions:**(b) What is the probability that the random variable will assume a value between 50 and 70? (Round your answer to three decimal places.)[Input box for answer](c) What is the probability that the random variable will assume a value between 45 and 75? (Round your answer to three decimal places.)[Input box for answer]---These questions require understanding and calculating the area under the normal distribution curve between specified values to determine probabilities. ### Understanding Normal DistributionA random variable is normally distributed with:- Mean ($\mu$) = 60- Standard deviation ($\sigma$) = 5#### (a) Normal Distribution CurveThe figure shows a normal distribution curve touching the horizontal axis at three standard deviations below and above the mean (at 45 and 75, respectively).##### Diagram Explanation:- **Normal Curve**: Bell-shaped curve representing data distribution.- **Mean ($\mu$)**: Center point of the curve ($\mu = 60$).- **Standard Deviations ($\sigma$)**: Measure of spread around the mean. Key points on the x-axis: - $\mu - 3\sigma$ (45) - $\mu - 2\sigma$ - $\mu - 1\sigma$ - $\mu$ (60) - $\mu + 1\sigma$ - $\mu + 2\sigma$ - $\mu + 3\sigma$ (75)- **Shaded Areas**: - **68.3%**: Within one standard deviation ($\mu \pm \sigma$). - **95.4%**: Within two standard deviations ($\mu \pm 2\sigma$). - **99.7%**: Within three standard deviations ($\mu \pm 3\sigma$).#### Sketching Normal CurvesSketch a normal curve for the probability density function. Label the horizontal axis with the following values: 45, 50, 55, 60, 65, 70, and 75.#### Additional Notes- **Three Graphs**: Each graph is identical, illustrating the normal distribution.- **Horizontal Axis Values**: Corresponding evenly across 45 to 75 on each curve.- **Conceptual Understanding**: Recognizing how data within +/- 1, 2, and 3 standard deviations account for a specific percentage of the distribution helps to elucidate the spread and behavior of the distribution.This overview and diagram aids in grasping the key aspects of normal distribution and how standard deviation defines data variability around the mean.

A random variable is normally distributed with a mean 60 and standard deviation 5. a) It is given…

Sketch a normal curve for the probability density function. Label the horizontal axis with values of 45, 50, 55, 60, 65, 70, and 75. 75 70 65 60 55 50 45 45 55 65 75 70 60 50 45 50 55 60 65 50 60 70 75 65 55 45 What is the probability the random variable will assume a value between 50 and 70? (Round your answer to three decimal places.) What is the probability the random variable will assume a value between 45 and 75? (Round your answer to three decimal places.)

Sketch a normal curve for the probability density function. Label the horizontal axis with values of 45, 50, 55, 60, 65, 70, and 75. 75 70 65 60 55 50 45 45 55 65 75 70 60 50 45 50 55 60 65 50 60 70 75 65 55 45 What is the probability the random variable will assume a value between 50 and 70? (Round your answer to three decimal places.) What is the probability the random variable will assume a value between 45 and 75? (Round your answer to three decimal places.)

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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Concept explainers

Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!

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**Transcription for Educational Content**

---

**Normal Distribution and Probability Density Functions**

In the figure above, we are presented with four graphs depicting different probability density functions modeled by normal curves. The task is to sketch a normal curve and label the horizontal axis with values of 45, 50, 55, 60, 65, 70, and 75.

Each graph displays a bell-shaped curve, characteristic of normal distribution, which is symmetrical around the mean.

**Graphs Explanation:**

1. **Graph 1:** The mean appears to be at 60, with the horizontal axis labeled from 75 to 45.

2. **Graph 2:** The mean appears to be at 65, with the horizontal axis labeled from 45 to 75.

3. **Graph 3:** The mean seems to be 60, with the axis values ranging from 45 to 75.

4. **Graph 4:** The mean looks to be at 65, with labels from 75 to 45.

The goal is to identify the correct arrangement of horizontal axis labels as stated in the task: 45, 50, 55, 60, 65, 70, and 75.

**Probability Questions:**

(b) What is the probability that the random variable will assume a value between 50 and 70? (Round your answer to three decimal places.)

[Input box for answer]

(c) What is the probability that the random variable will assume a value between 45 and 75? (Round your answer to three decimal places.)

[Input box for answer]

---

These questions require understanding and calculating the area under the normal distribution curve between specified values to determine probabilities.

### Understanding Normal Distribution

A random variable is normally distributed with:
- Mean ($\mu$) = 60
- Standard deviation ($\sigma$) = 5

#### (a) Normal Distribution Curve

The figure shows a normal distribution curve touching the horizontal axis at three standard deviations below and above the mean (at 45 and 75, respectively).

##### Diagram Explanation:

- **Normal Curve**: Bell-shaped curve representing data distribution.
- **Mean ($\mu$)**: Center point of the curve ($\mu = 60$).
- **Standard Deviations ($\sigma$)**: Measure of spread around the mean. Key points on the x-axis:
- $\mu - 3\sigma$ (45)
- $\mu - 2\sigma$
- $\mu - 1\sigma$
- $\mu$ (60)
- $\mu + 1\sigma$
- $\mu + 2\sigma$
- $\mu + 3\sigma$ (75)
- **Shaded Areas**:
- **68.3%**: Within one standard deviation ($\mu \pm \sigma$).
- **95.4%**: Within two standard deviations ($\mu \pm 2\sigma$).
- **99.7%**: Within three standard deviations ($\mu \pm 3\sigma$).

#### Sketching Normal Curves

Sketch a normal curve for the probability density function. Label the horizontal axis with the following values: 45, 50, 55, 60, 65, 70, and 75.

#### Additional Notes

- **Three Graphs**: Each graph is identical, illustrating the normal distribution.
- **Horizontal Axis Values**: Corresponding evenly across 45 to 75 on each curve.
- **Conceptual Understanding**: Recognizing how data within +/- 1, 2, and 3 standard deviations account for a specific percentage of the distribution helps to elucidate the spread and behavior of the distribution.

This overview and diagram aids in grasping the key aspects of normal distribution and how standard deviation defines data variability around the mean.

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Oxygen demand is a term biologists use to describe the oxygen needed by fish and other aquatic organisms for survival. The Environmental Protection Agency conducted a study of a wetland area. In this wetland environment, the mean oxygen demand was μ = 9.9 mg/L with 95% of the data ranging from 6.3 mg/L to 13.5 mg/L. Let x be a random variable that represents oxygen demand in this wetland environment. Assume x has a probability distribution that is approximately normal. (a) Use the 95% data range to estimate the standard deviation for oxygen demand. (b) An oxygen demand below 8 indicates that some organisms in the wetland environment may be dying. What is the probability that the oxygen demand will fall below 8 mg/L? (Round your answer to four decimal places.) (c) A high oxygen demand can also indicate trouble. An oxygen demand above 12 may indicate an overabundance of organisms that endanger some types of plant life. What is the probability that the oxygen demand will exceed 12…