Question

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.

Short answer

Distribution of proper notation$X~U(0,14)$

The value of the theoretical mean is $7$.

The value of standard deviation is $4.04$

Step-by-step solution

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.

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$X~U(0,14)$

Calculate the theoretical mean 

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

Place the value of $aandb$

Therefore,

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

$=\frac{14}{2}$

$=7$

Calculate the standard deviation 

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

Place the value of $aandb$

Now,

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

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

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

$\approx 4.04$