Chapter 10.3, Problem 13PPS

a.

To determine

To calculate and compare mean and standard deviation of the movies.

Answer to Problem 13PPS

Movie A mean is 7.19 and standard deviation is 0.807

Movie B mean is 6.75 and standard deviation is 2.88

Explanation of Solution

Given information:

Movie A Data: 7,8,7,6,8,6,7,8,6,8,8,6,7,7,8,8

Movie B data: 9,5,10,6,3,10,9,4,8,3,9,9,2,8,10,3

Formula used: Mean formula is,

  $\overline{X}=\frac{{\displaystyle \sum fX}}{N}$

Where, f is frequency

N is total number of items

Standard deviation is,

  $SD=\sqrt{\frac{{\displaystyle \sum }{(x-\overline{x})}^2}{n}}$

Where, SD= standard deviation,

x= value

Calculation:

For Movie A,

Substituting the values in mean formula,

  $\overline{x}=\frac{7+8+7+6+8+6+7+8+6+8+8+6+7+7+8+8}{16}$

So,

  $\overline{x}=\frac{115}{16}$

On solving,

  $\overline{x}=7.19$

Hence, Mean is 7.19

Value	Difference	Square of difference
7	-0.19	0.03
8	0.81	0.65
7	-0.19	0.03
6	-1.19	1.41
8	0.81	0.65
6	-1.19	1.41
7	-0.19	0.03
8	0.81	0.65
6	-1.19	1.41
8	0.81	0.65
8	0.81	0.65
6	-1.19	1.41
7	-0.19	0.03
7	-0.19	0.03
8	0.81	0.65
8	0.81	0.65
Total		10.34

Standard deviation,

Substituting the values in standard deviation formula,

  $\sigma =\sqrt{\frac{10.43}{16}}$

On solving,

  $\sigma$ =0.807

Standard deviation is 0.807

For Movie B,

Substituting the values in mean formula,

  $\overline{x}=\frac{9+5+10+6+3+10+9+4+8+3+9+9+2+8+10+3}{16}$

So,

  $\overline{x}=\frac{108}{16}$

On solving,

  $\overline{x}=6.75$

Hence, Mean is 6.75

Value	Difference	Square of difference
9	2..25	5.06
5	-1.75	3.06
10	3.25	10.56
6	-0.75	0.56
3	-3.75	14.06
10	3.25	10.56
9	2.25	5.06
4	-2.75	7.56
8	1.75	1.56
3	-3.75	14.06
9	2.25	5.062
9	2.25	5.06
2	-4.75	22.56
8	1.25	1.56
10	3.25	10.56
3	-3.75	14.06
Total		132.5

For standard deviation,

Substituting the values in standard deviation formula,

  $SD=\sqrt{\frac{132.5}{16}}$

So,

  $SD=\sqrt{8.28}$

On solving,

  $SD=2.88$

Standard deviation is 2.88

Comparing the mean, the average rating of Movie A is better than movie B.

On comparing standard deviation, movie A has a lower standard deviation that means the points are closer to mean and movie B have higher standard deviation that means the points are spread out over large range of values.

b.

To determine

To evaluate why movie A is better, and also why movie B is better?

Explanation of Solution

Given information: Movie A mean is 7.19 and standard deviation is 0.807

Movie B mean is 6.75 and standard deviation is 2.88

Formula used: Comparison of mean and standard deviation.

Evaluation:

For movie A:

Movie A is better than movie B as it has a higher average rating than movie B. this implies, that movie A is liked by more people and is a better movie according to the mean rating given to it by the people.

For movie B:

Movie B is better than movie A because the standard deviation of movie B is higher that means that points are spread over large range of values. Hence there is a mixed reviews about it and the viewer will find movie B better, due to its genre.