The number of degrees of freedom associated with the chi-square distribution in a test of independence is a. number of sample items minus 1. b. number of populations minus number of estimated parameters minus 1. c. number of populations minus 1. d. number of rows minus 1 times number of columns minus 1.

d) the number of rows minus 1 times number of columns minus 1

The degrees of freedom γ= (r-1)(s-1)

Step-by-step explanation:

chi-square distribution:-

In this chi-square test , we test if two attributes A  and B under consideration are independent or not .

here we will choose null hypothesis (H₀): Attributes are independent

Degrees of freedom :  γ= (r-1)(s-1)

where r = number of rows

and 's' = number of columns

chi - squared test

Χ² = ∑(O-E)²/ E

'O' be the observed frequency

'E' be the Expected frequency

pentagon

step-by-step explanation:

the coordinates   a(5, -2), b(3,-1),c(-4,-4) d(-3,8) and e(-1,4) forms the vertices of the polygon when they are connected in order.

in order to classify that which polygon is it with the given coordinates, first count the number of vertices and then count the number of sides it can make with these vertices.

since, the given coordinates have five vertices, therefore, it will make five sides. moreover, pentagon is regular in nature therefore it will be convex.

now, a polygon with five sides is known as pentagon, therefore, the given polygon is the pentagon.