Chapter 10.5, Problem 5CYU

a.

To determine

To Graph: To construct a histogram for each set of data.

Explanation of Solution

Given information: Given data is,

K’s distance: 17.2, 18.28, 17.28, 17.36, 18.08, 17.43, 17.71, 17.46, 18.26, 17.51, 17.58, 17.41, 18.21, 17.34, 17.63, 17.55, 17.26, 17.18, 17.78, 17.51, 17.83, 17.92, 18.04, 17.91

M’s distance: 18.88, 19.24, 17.63, 18.96, 18.19, 18.21, 18.46, 17.47, 18.49, 17.86,

18.93, 18.73, 18.34, 18.67, 18.56, 18. 79, 18.47, 18.84, 17.94, 18.7

Graph:

For K’s Distances

  

For M’s Distances

  

Interpretation:

Both the graphs are unstable. Values are increasing and decreasing.

b.

To determine

To describe the center and spread the data using five-number summary.

Answer to Problem 5CYU

The K’s Distances five number summary is,

Minimum=17.18

Lower quartile=17.38

Median=17.58

Upper quartile=18

Maximum=18.56

The M’s Distances five number summary is,

Minimum=17.74

Lower quartile=18.06.

Median=18.56

Upper quartile=18.85

Maximum=19.24

Explanation of Solution

Given information: Given data is,

K’s distance: 17.2, 18.28, 17.28, 17.36, 18.08, 17.43, 17.71, 17.46, 18.26, 17.51, 17.58, 17.41, 18.21, 17.34, 17.63, 17.55, 17.26, 17.18, 17.78, 17.51, 17.83, 17.92, 18.04, 17.91

M’s distance: 18.88, 19.24, 17.63, 18.96, 18.19, 18.21, 18.46, 17.47, 18.49, 17.86,

18.93, 18.73, 18.34, 18.67, 18.56, 18. 79, 18.47, 18.84, 17.94, 18.7

Formula used: Median is,

  $M=\frac{{(\frac{n}{2})}^{th}+{(\frac{n}{2}+1)}^{th}}{2}term$ ,

Lower quartile is,

  $Q1=(n+1)\frac{1}{4}$

Upper quartile is,

  $Q3=(n+1)\frac{3}{4}$

Where, n is total number of terms.

Calculation:

Re-arranging the data in an ascending order,

K’s Distances

17.18,17.2,17.26,17.28,17.34,17.36,17.41,17.43,17.46,17.51,17.51,17.55,17.58,17.63,17.71,17.78,17.83,17.91,17.92,18.04,18.08,18.21,18.26,18.28,18.56

M’s Distances

17.74,17.47,17.63,17.86,17.92,17.94,18.19,18.21,18.34,18.46,18.47,18.49,18.56,18.61,18.67,18.7,18.73,18.79,18.84,18.87,18.88,18.93,18.96,19.18,19.24,

For K’s Distances

For median,

Substituting values,

  $M=\frac{25+1^{th}}{2}term$

On solving,

  $M={13}^{th}term$

Substituting the value,

  $M=17.58$

Hence, Median is 17.58

For lower quartile,

Substituting the values,

  $Q1=(25+1)\frac{1}{4}$

On solving,

  $Q1=6.5$

Hence, adding 6^th and 7^th to get the answer,

  $Q1=\frac{17.36+17.41}{2}$

On solving,

  $Q1=17.38$

Hence, Lower quartile is 17.38

For higher quartile,

Substituting the values,

  $Q3=(25+1)\frac{3}{4}$

Onn solving,

  $Q3=19.5$

Hence, adding 19^th and 20^th to get the answer,

  $Q3=\frac{17.96+18.04}{2}$

On solving,

  $Q3=18$

Hence, higher quartile is 18

Minimum is the smallest value of dataset.

Hence, minimum is 17.18.

Maximum is the largest value of the dataset.

Hence, maximum is 18.56.

The five number summary is,

17.18, 17.38, 17.58, 18, 18.56

For M’s Distances

For median,

Substituting the values,

  $M=\frac{25+1^{th}}{2}term$

On solving,

  $M={13}^{th}term$

Substituting the value,

  $M=18.56$

Hence, Median is 18.56

For lower quartile,

Substituting the values,

  $Q1=(25+1)\frac{1}{4}$

On solving,

  $Q1=6.5$

Hence, adding 6^th and 7^th to get the answer,

  $Q1=\frac{17.94+18.19}{2}$

On solving,

  $Q1=18.06$

Hence, Lower quartile is 18.06

For higher quartile,

Substituting the values,

  $Q3=(25+1)\frac{3}{4}$

On solving,

  $Q3=19.5$

Hence, adding 19^th and 20^th to get the answer,

  $Q3=\frac{18.84+18.87}{2}$

On solving,

  $Q3=18.85$

Hence, higher quartile is 18.85

Minimum is the smallest value of dataset.

Hence, minimum= 17.74.

Maximum is the largest value of the dataset.

Hence, maximum= 19.24.

The five number summary is,

17.74, 18.06, 18.56, 18.85, 19.24

Five number summary is used, since, it is less time consuming and more efficient.