Chapter 10.5, Problem 20PPS

i.

To determine

To find mean median mode, range and standard deviation.

Answer to Problem 20PPS

Mean=18.68

Median=15.51

Mode=no mode

Range=40.06

Standard deviation=11.62

Explanation of Solution

Given information:

Cost($): 14.98, 24.61, 12.84, 16.05, 42.80, 2.14, 19.26

Formula used:

Mean= $\frac{{\displaystyle \sum _{i=1}^n{X}_i}}{n}$

Where, n=total number of terms,

  ${\displaystyle \sum _{i=1}^n{X}_i}$ = sum of all terms

Mode= most repeating numbers.

Median= $\frac{{(n+1)}^{th}}{2}observation$

Range= maximum- minimum

Standard deviation, ${\sigma }_{}=\sqrt{\frac{{\displaystyle \sum _{}^{}{(x_i^{}-\mu )}^2}}{N}}$

Where $x_i^{}$ = each value from data

  $\mu$ =mean of data

  $N$ = size of data

Calculation:

Substituting the value,

Mean= $\frac{14.98+24.61+12.84+16.05+42.80+2.14+19.26}{7}=\frac{132.68}{7}=18.68$

Sort the number in ascending order,

2.14, 12.84, 14.98, 16.05, 19.26, 24.61, 42.8

Median= $\frac{{(\frac{N}{2})}^{th}ob+{(\frac{N}{2}+1)}^{th}ob}{2}=\frac{{(\frac{8}{2})}^{th}+{(\frac{8}{2}+1)}^{th}}{2}=\frac{4^{th}+5^{th}}{2}=\frac{14.98+16.05}{2}=15.51$

Mode= no mode.

Range= $\max -\min =42.8-2.14=40.06$

For standard deviation,

Values	Difference= $x_{}^{}-\mu$	( $x_{}^{}-\mu {)}^2$
14.98	-3.97	15.79
24.61	5.65	31.98
12.84	-6.11	37.38
16.05	-2.90	8.43
42.80	23.84	568.61
2.14	-16.81	282.72
19.26	0.30	0.09

  ${\displaystyle \sum _{}^{}{(x_{}^{}-\mu )}^2}$ =945.033

  ${\sigma }_{}=\sqrt{\frac{{\displaystyle \sum _{}^{}{(x_i^{}-\mu )}^2}}{N}}$ = ${\sigma }_{}=\sqrt{\frac{945.03}{7}=}11.61$

ii.

To determine

To find mean median mode, range and standard deviation.

Answer to Problem 20PPS

Mean=17.62

Median=14.92

Mode=no mode

Range=37.8

Standard deviation=10.80

Explanation of Solution

Given information:

Sales tax: 7%

Cost ($): 14.98, 24.61, 12.84, 16.05, 42.80, 2.14, 19.26

Formula used:

Mean= $\frac{{\displaystyle \sum _{i=1}^n{X}_i}}{n}$

Where, n=total number of terms,

  ${\displaystyle \sum _{i=1}^n{X}_i}$ = sum of all terms

Mode= most repeating numbers.

Median= $\frac{{(n+1)}^{th}}{2}observation$

Range= maximum- minimum

Standard deviation, ${\sigma }_{}=\sqrt{\frac{{\displaystyle \sum _{}^{}{(x_i^{}-\mu )}^2}}{N}}$

Where $x_i^{}$ = each value from data

  $\mu$ =mean of data

  $N$ = size of data

Calculation:

After removing sales tax,

New data is 13.93, 22.88, 11.94, 14.92, 39.80, 1.99, and 17.91

Substituting the value,

Mean= $\frac{13.93+22.88+11.94+14.92+39.80+1.99+17.91}{7}=\frac{123.37}{7}=17.62$

Sort the number in ascending order,

1.99,11.94,13.93,14.92,17.91,22.88,39.8

Median= ${(\frac{N+1}{2})}^{th}ob$

  ${(\frac{7+1}{2})}^{th}$ =4^th ob=14.92

Mode= no mode

Range= $\max -\min =39.8-1.99=37.81$

For standard deviation,

1.99,11.94,13.93,14.92,17.91,22.88 39.8

Values	Difference= $x_{}^{}-\mu$	( $x_{}^{}-\mu {)}^2$
1.99	-15.63	244.43
11.94	-5.68	32.311
13.93	-3.69	13.64
14.92	-2.70	7.31
17.91	0.28	0.081
22.88	5.25	27.62
39.8	22.17	491.76

  ${\displaystyle \sum _{}^{}{(x_{}^{}-\mu )}^2}$ =817.16

  ${\sigma }_{}=\sqrt{\frac{{\displaystyle \sum _{}^{}{(x_i^{}-\mu )}^2}}{N}}$ = ${\sigma }_{}=\sqrt{\frac{817.16}{7}=}10.80$