Assume that Math test scores are normally distributed with a mean of 200 and a standard deviation of 40. If you were to score 120 on this exam what percentage of those taking the test scored below are you?What percentage of those taken the test scored below you?(Round to the nearest tenth as needed.) ### Section 14.4 – The Normal DistributionThe normal distribution (normal curve) is a way to study data using the information you have learned about. A normal distribution is a bell-shaped curve that is symmetric with respect to the mean. A normal distribution usually applies to a whole population, not a sample, therefore the mean is now called \( \mu \) (mu).#### Diagram Explanation:The diagram illustrates the bell-shaped curve of a Normal Distribution. - The curve is symmetric about the mean \( \mu \).- The horizontal axis represents the range of data values, marked in units of standard deviation (SD) from the mean \( \mu \).- The curve is divided into segments based on standard deviations from the mean: \( -3\sigma \), \( -2\sigma \), \( -1\sigma \), \( \mu \), \( +1\sigma \), \( +2\sigma \), and \( +3\sigma \).#### Key Segments of the Curve:- The area between \( -1\sigma \) and \( +1\sigma \) contains approximately 68% of the data.- The area between \( -2\sigma \) and \( +2\sigma \) contains approximately 95% of the data.- The area between \( -3\sigma \) and \( +3\sigma \) contains approximately 99.7% of the data.#### Percentages Representing the Data in Each Segment:- \( \mu \pm 1\sigma \): 34.0% (each side)- \( \mu \pm 2\sigma \): 13.5% (each side)- \( \mu \pm 3\sigma \): 2.35% (each side)Outside of \( \pm 3\sigma \):- Less than 0.15% of the data lies beyond \( \pm 3\sigma \).### Key Takeaways:- The normal distribution is fundamental in statistics because of its unique properties.- Understanding the percentages within each standard deviation range helps in predicting the distribution of data within any given dataset. - This knowledge is essential for tasks such as quality control, standard deviation analysis, and various applications in psychology, finance, and other fields that rely on statistical data.This section provides a foundational understanding of how to evaluate and interpret data using the principles of the normal distribution. ### Z-Score TableA Z-score (or standard score) indicates how many standard deviations an element is from the mean. Here is the Z-score table which provides the cumulative probability of a standard normal distribution. This table helps in finding the probability that a statistic is observed below, above, or between values on the standard normal distribution. #### How to Read the Z-Score TableEach number in the table represents the probability that a value in a standard normal distribution will fall to the left of the corresponding Z-score. To find the probability of a Z-score:1. Identify the Z-score you are looking for. For example, if you need the probability up to a Z-score of 0.56, find the row for 0.5 and then move over to the column for 0.06.2. The value where the row and column intersect is the probability.Here is the provided Z-score table:| z | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.06 | 0.07 | 0.08 | 0.09 ||-----|--------|--------|--------|--------|--------|--------|--------|--------|--------|| 0.0 | 0.50000 | 0.50399 | 0.50798 | 0.51197 | 0.51595 | 0.51994 | 0.52392 | 0.52790 | 0.53188 || +0.1 | 0.53983 | 0.54380 | 0.54776 | 0.55172 | 0.55567 | 0.55962 | 0.56356 | 0.56749 | 0.57142 || +0.2 | 0.57926 | 0.58317 | 0.58706 | 0.59095 | 0.59483 | 0.59871 | 0.60257 | 0.60642 | 0.61026 || +0.3 | 0.61791 | 0.62172 | 0.62552 | 0.62930 | 0.63307 | 0.63683 | 0.64058 | 0.64431

The provided information is The mean and standard deviation is µ=200 σ=40 P(X<120)=?…

Answered: Assume that Math test scores are…

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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Assume that Math test scores are normally distributed with a mean of 200 and a standard deviation of 40. If you were to score 120 on this exam what percentage of those taking the test scored below are you? What percentage of those taken the test scored below you? (Round to the nearest tenth as needed.)

### Section 14.4 – The Normal Distribution

The normal distribution (normal curve) is a way to study data using the information you have learned about. A normal distribution is a bell-shaped curve that is symmetric with respect to the mean. A normal distribution usually applies to a whole population, not a sample, therefore the mean is now called \( \mu \) (mu).

#### Diagram Explanation:
The diagram illustrates the bell-shaped curve of a Normal Distribution.

- The curve is symmetric about the mean \( \mu \).
- The horizontal axis represents the range of data values, marked in units of standard deviation (SD) from the mean \( \mu \).
- The curve is divided into segments based on standard deviations from the mean: \( -3\sigma \), \( -2\sigma \), \( -1\sigma \), \( \mu \), \( +1\sigma \), \( +2\sigma \), and \( +3\sigma \).

#### Key Segments of the Curve:
- The area between \( -1\sigma \) and \( +1\sigma \) contains approximately 68% of the data.
- The area between \( -2\sigma \) and \( +2\sigma \) contains approximately 95% of the data.
- The area between \( -3\sigma \) and \( +3\sigma \) contains approximately 99.7% of the data.

#### Percentages Representing the Data in Each Segment:
- \( \mu \pm 1\sigma \): 34.0% (each side)
- \( \mu \pm 2\sigma \): 13.5% (each side)
- \( \mu \pm 3\sigma \): 2.35% (each side)

Outside of \( \pm 3\sigma \):
- Less than 0.15% of the data lies beyond \( \pm 3\sigma \).

### Key Takeaways:
- The normal distribution is fundamental in statistics because of its unique properties.
- Understanding the percentages within each standard deviation range helps in predicting the distribution of data within any given dataset.
- This knowledge is essential for tasks such as quality control, standard deviation analysis, and various applications in psychology, finance, and other fields that rely on statistical data.

This section provides a foundational understanding of how to evaluate and interpret data using the principles of the normal distribution.

### Z-Score Table

A Z-score (or standard score) indicates how many standard deviations an element is from the mean. Here is the Z-score table which provides the cumulative probability of a standard normal distribution. This table helps in finding the probability that a statistic is observed below, above, or between values on the standard normal distribution.

#### How to Read the Z-Score Table

Each number in the table represents the probability that a value in a standard normal distribution will fall to the left of the corresponding Z-score. To find the probability of a Z-score:

1. Identify the Z-score you are looking for. For example, if you need the probability up to a Z-score of 0.56, find the row for 0.5 and then move over to the column for 0.06.
2. The value where the row and column intersect is the probability.

Here is the provided Z-score table:

| z | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.06 | 0.07 | 0.08 | 0.09 |
|-----|--------|--------|--------|--------|--------|--------|--------|--------|--------|
| 0.0 | 0.50000 | 0.50399 | 0.50798 | 0.51197 | 0.51595 | 0.51994 | 0.52392 | 0.52790 | 0.53188 |
| +0.1 | 0.53983 | 0.54380 | 0.54776 | 0.55172 | 0.55567 | 0.55962 | 0.56356 | 0.56749 | 0.57142 |
| +0.2 | 0.57926 | 0.58317 | 0.58706 | 0.59095 | 0.59483 | 0.59871 | 0.60257 | 0.60642 | 0.61026 |
| +0.3 | 0.61791 | 0.62172 | 0.62552 | 0.62930 | 0.63307 | 0.63683 | 0.64058 | 0.64431

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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