# Understanding a Probability Distribution for Book RatingsThe histogram shows the reviewer ratings on a scale from 1 (lowest) to 5 (highest) of a recently published book.**Tasks:**1. Find the mean, variance, and standard deviation of the probability distribution.2. Interpret the results.## Histogram ExplanationThe histogram provided is a graphical representation of the reviewer ratings with the following probability distribution:- **Rating 1:** Probability 0.022- **Rating 2:** Probability 0.087- **Rating 3:** Probability 0.211- **Rating 4:** Probability 0.232- **Rating 5:** Probability 0.448The histogram shows the frequencies of the ratings, with the height of each bar corresponding to the probability of each rating. ## Solving the Tasks### (a) Calculations#### MeanThe mean of the probability distribution is calculated as follows:\[ \text{Mean} (\mu) = \sum (X \cdot P(X)) \]#### VarianceThe variance of the probability distribution is calculated as follows:\[ \text{Variance} (\sigma^2) = \sum [(X - \mu)^2 \cdot P(X)] \]#### Standard DeviationThe standard deviation is the square root of the variance:\[ \text{Standard Deviation} (\sigma) = \sqrt{\sigma^2} \]### (b) InterpretationOptions are provided to help interpret the results:- **A.** The average rating for the book is approximately 4.- **B.** The average rating for the book is approximately 1.- **C.** The typical rating deviates from the mean by about 4.- **D.** The typical rating deviates from the mean by about 1.Please complete the calculations to find the mean, variance, and standard deviation to interpret the results accurately.**Select your answer(s):**(Interactive buttons for selecting answers)This exercise helps in understanding how to find and interpret key statistical measures for a given probability distribution.

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Description: # Understanding a Probability Distribution for Book RatingsThe histogram shows the reviewer ratings on a scale from 1 (lowest) to 5 (highest) of a recently published book.**Tasks:**1. Find the mean, variance, and standard deviation of the probabilit

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The histogram shows the reviewer ratings on a scale from 1 (lowest) to 5 (highest) of a recently published book. (a) Find the mean, variance, and standard deviation of the probability distribution. (b) Interpret the results. (a) The mean is (Type an integer or a decimal. Do not round.) The variance is.

The histogram shows the reviewer ratings on a scale from 1 (lowest) to 5 (highest) of a recently published book. (a) Find the mean, variance, and standard deviation of the probability distribution. (b) Interpret the results. (a) The mean is (Type an integer or a decimal. Do not round.) The variance is.

MATLAB: An Introduction with Applications

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Author:Amos Gilat

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Concept explainers

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.

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Question

# Understanding a Probability Distribution for Book Ratings

The histogram shows the reviewer ratings on a scale from 1 (lowest) to 5 (highest) of a recently published book.

**Tasks:**
1. Find the mean, variance, and standard deviation of the probability distribution.
2. Interpret the results.

## Histogram Explanation
The histogram provided is a graphical representation of the reviewer ratings with the following probability distribution:

- **Rating 1:** Probability 0.022
- **Rating 2:** Probability 0.087
- **Rating 3:** Probability 0.211
- **Rating 4:** Probability 0.232
- **Rating 5:** Probability 0.448

The histogram shows the frequencies of the ratings, with the height of each bar corresponding to the probability of each rating.

## Solving the Tasks

### (a) Calculations
#### Mean
The mean of the probability distribution is calculated as follows:

\[ \text{Mean} (\mu) = \sum (X \cdot P(X)) \]

#### Variance
The variance of the probability distribution is calculated as follows:

\[ \text{Variance} (\sigma^2) = \sum [(X - \mu)^2 \cdot P(X)] \]

#### Standard Deviation
The standard deviation is the square root of the variance:

\[ \text{Standard Deviation} (\sigma) = \sqrt{\sigma^2} \]

### (b) Interpretation
Options are provided to help interpret the results:

- **A.** The average rating for the book is approximately 4.
- **B.** The average rating for the book is approximately 1.
- **C.** The typical rating deviates from the mean by about 4.
- **D.** The typical rating deviates from the mean by about 1.

Please complete the calculations to find the mean, variance, and standard deviation to interpret the results accurately.

**Select your answer(s):**

(Interactive buttons for selecting answers)

This exercise helps in understanding how to find and interpret key statistical measures for a given probability distribution.

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