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5.2

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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" $=\sqrt{16.33}$

$\approx 4.04$