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Expert-verified Found in: Page 318 ### Introductory Statistics

Book edition OER 2018
Author(s) Barbara Illowsky, Susan Dean
Pages 902 pages
ISBN 9781938168208 # The data that follow are the number of passengers on 35 different charter fishing boats. The sample mean = 7.9 and the sample standard deviation = 4.33. The data follow a uniform distribution where all values between and including zero and 14 are equally likely. State the values of $a$ and $b$. Write the distribution in proper notation, and calculate the theoretical mean and standard deviation. Distribution of proper notation $X~U\left(0,14\right)$

The value of the theoretical mean is $7$.

The value of standard deviation is $4.04$

See the step by step solution

## Step 1: Given Information

Given in the question that a table with the number of passengers on $35$ different charter fishing boats.

The sample mean $=$$7.9$ and the sample standard deviation $=4.33$.

We need to determine the values of $a$ and $b$

We have to write the distribution in proper notation, and calculate the theoretical mean and standard deviation.

## Step 2: Vales of a and b

Let's consider the above table

From the table , the lowest values is $0$and the highest value is $14$.

Therefore, the variable $a=0$

$b=14$

Hence, the provided data follows the uniform distribution such that $X~U\left(0,14\right)$

## Step 3: Calculate the theoretical mean

The theoretical mean of the uniform distribution is $\mu =\frac{a+b}{2}$

Place the value of $aandb$

Therefore,

$\mu =\frac{0+14}{2}$

$=\frac{14}{2}$

$=7$

## Step 4: Calculate the standard deviation

The standard deviation of the uniform distribution is $\sigma =\sqrt{\frac{\left(b-a{\right)}^{2}}{12}}$

Place the value of $aandb$

Now,

$\sigma =\sqrt{\frac{\left(14-0{\right)}^{2}}{12}}$

$=\sqrt{\frac{196}{12}}$

localid="1647957978309" $=\sqrt{16.33}$

$\approx 4.04$ ### Want to see more solutions like these? 