Question

A sociologist randomly selects single adults for different groups of four, and the random variable x is the number of adults in the group who say that the most fun way to flirt is in person. (Based on a Messaging Survey)

x	p(x)
0	0.092
1	0.159
2	0.357
3	0.320
4	0.072

Answer the following questions and make sure you apply the round-off rule and units correctly:

(a) Does the table describe a probability distribution? Why yes or why no?

(b) Find the mean

(c) Find the variance

(d) Find the standard deviation

(e) Use the range rule of thumb to identify the range of usual values for the number of adults who agree the sociologist's statement.

(f) Is it unusual to get three adults who agree the sociologist's statement among four randomly selected adults? Give your conclusion.

Answer 1

a. Here all values of p( x ) are positive.

Σ P ( x ) = P(0)+P(1)+P(2)+P(3)+P(4)

= 0.092+0.159+0.357+0.320+0.072

=1

Therefore Σ P ( x ) = 1

Here both conditions are satisfied.

Therefore the given table describe a probability distribution.

To calculate mean and variance we have to prepare a table as follows.

Xi	P ( Xi​​​​​​ )	Xi•P(Xi )	Xi 2•P ( Xi )
0	0.092	0	0
1	0.159	0.159	0.159
2	0.357	0.714	1.428
3	0.320	0.960	2.880
4	0.072	0.288	1.152
Total	Σ= 1	Σ= 2.121	Σ= 5.619

b.

Mean = E (X) = Σ X_i • P ( X_i ) =

= 2.121

Therefore  mean is 2.121

c.

Variance = Σ X_i ² • P ( X_i ) - [ Σ X_i • P ( X_i ) ]²

= 5.619 - ( 2.121 )²

= 5.619 - 4.498641

= 1.120359

= 1.1204

Therefore variance is 1.1204

d.

The standard deviation =

=

= 1.05848949

= 1.0585

Therefore standard deviation is 1.0585