Question

250 employees in an organization were surveyed, and the following data was collected about their hair color and height: Which inference can be made from this two-way table? A. Being above 180 centimeters and having black hair are independent of each other. B. Being above 180 centimeters is dependent on having black hair. C. Having black hair is dependent on being above 180 centimeters. D. Being above 180 centimeters and having black hair are the same.

Hair
Color Height
Less than
175 cm 175-180 cm Above 180 cm Total
Black 38 32 30 100
Brown 27 24 19 70
Blonde 21 33 26 80
Total 86 89 75 250

Answer 1

Option: A is the correct answer.

A. Being above 180 centimeters and having black hair are independent of each other.

Explanation:

Height : Less than 175 cm 175-180 cm Above 180 cm Total

Hair Color

Black 38 32 30 100

Brown 27 24 19 70

Blonde 21 33 26 80

Total 86 89 75 250

Two events A and B are said to be independent if:

`P(A\bigcap B)=P(A)* P(B)`

else they are dependent.

A)

Being above 180 centimeters and having black hair are independent of each other.

Let A denote Black hair

and B denote above 180 cm.

`P(A)=(100)/(250)=(10)/(25)`

and
`P(B)=(75)/(250)=(3)/(10)`

This means that:

`P(A)* P(B)=(10)/(25)* (3)/(10)=(3)/(25)`

Also,

`P(A\bigcap B)=(30)/(250)=(3)/(25)`

Since,

`P(A\bigcap B)=P(A)* P(B)`

Hence, events A and events B are independent.

Hence, option: A is correct.

Answer 2

A. Being above 180 centimeters and having black hair are independent of each other.

Explanation:

These are instinctively two characteristics that are not related and the table data proves it.

If they were dependent, you would have only people with black hair above 180 cm, and all people with black hair would be above 180 cm.

You can be above 180 cm and have black, brown or blonde hair. The table shows a proportion of people > 180 cm about equal (around 30% of the sampling) for each hair color.

And people above 180 cm only represent 30% of the people with black hair.