Question

A sample of 100 cars driving on a freeway during a morning commute was drawn, and the number of occupants in each car was recorded. The results were as follows: NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Occupants 1 2 3 4 5 Number of Cars 74 10 11 3 2 Find the sample standard deviation of the number of occupants. The sample standard deviation is 37.60 37.60 Incorrect . (Round the final answer to two decimal places.)

Answer 1

`E(X)=1*0.74 +2*0.1 +3*0.11+ 4*0.03 +5*0.02=1.49`

`Var(X)=E(X^2)-[E(X)]^2 =3.11-(1.49)^2 =0.8899`

`Sd(X)=√(Var(X))=√(0.8899)=0.943`

Explanation:

For this case we have the following data given:

X 1 2 3 4 5

F 74 10 11 3 2

The total number of values are 100, so then we can find the empirical probability dividing the frequency by 100 and we got the followin distribution:

X 1 2 3 4 5

P(X) 0.74 0.10 0.11 0.03 0.02

Previous concepts

In statistics and probability analysis, the expected value "is calculated by multiplying each of the possible outcomes by the likelihood each outcome will occur and then summing all of those values".

The variance of a random variable Var(X) is the expected value of the squared deviation from the mean of X, E(X).

And the standard deviation of a random variable X is just the square root of the variance.

Solution to the problem

In order to calculate the expected value we can use the following formula:

`E(X)=\sum_(i=1)^n X_i P(X_i)`

And if we use the values obtained we got:

`E(X)=1*0.74 +2*0.1 +3*0.11+ 4*0.03 +5*0.02=1.49`

In order to find the standard deviation we need to find first the second moment, given by :

`E(X^2)=\sum_(i=1)^n X^2_i P(X_i)`

And using the formula we got:

`E(X^2)=1^2 *0.74 +2^2 *0.1 +3^2 *0.11 +4^2 0.03 +5^2 *0.02=3.11`

Then we can find the variance with the following formula:

`Var(X)=E(X^2)-[E(X)]^2 =3.11-(1.49)^2 =0.8899`

And then the standard deviation would be given by:

`Sd(X)=√(Var(X))=√(0.8899)=0.943`