MAT 209 - Homework #2 Histogram Drawing and Data of Brain Volume

Homework Assignment #2, due Wednesday, February 5, 2019

1.      Visit www.pbialas.com website and click on HISTOGRAM 1 link; (Frequency Distribution Histogram - Histogram of Brain Volumes – Model Problem/Tutorial)

2.     Use Brain Volume Data to answer Preliminaries, (W’s)

3.     In Graphical Analysis respond in writing to (1), (2), and (3)

4.     Construct frequency distribution table for Brain Volume Data using [900,1000), [1000,1100), … [1400,1500) classes.

5.     Construct a histogram of Brain Volumes, label the axis, and attach meaningful title to your display.

6.     In   Extension section of the page answer (1), (Construct relative frequency distribution table for Brain Volume Data using [900,1000), [1000,1100), … [1400,1500) classes.

7.     Construct a relative frequency histogram of Brain Volumes, label the axis, and attach meaningful title to your display.

Note: No electronic submission, please. COMPLETE - 2/5/2019

Frequency Distribution Histogram: Histogram of Brain Volumes-Model Problem/Tutorial

Brain Volume Data

1005,963,1035,1027,1281,1272,1057,1079,1034,1070,1173,1079,1067,1104,1347,1439,1029,1100,1204,1160

Statistical “W”s

http://n.neurology.org/content/50/5/1246

Who: Monozygotic twins

What: Brain size in volume, cm³

When: 1998

Where: N/A

How: Brain size determined through MRI and quantitative image analysis.

Graphical Analysis

1) Skewed right = right tail. There are higher frequency values on the left side than the right side.

2) Center of distribution estimate = 1050 cm³, because the highest frequency is between 1000-1100. The lower-end of the center of distribution is 1000 cm³ and the higher-end of center 1300 cm³ due to the shape of the histogram.

3) Outliers are the single frequency values such as 963, 1347, and 1439. Together they make up the single frequencies in the brackets 900-1000, 1300-1400, and 1400-1500.

Stem & Leaf Display

9	63
10	05	27	29	34	35	57	67	70	79	79
11	00	04	60	73
12	04	72	81
13	47
14	39
15

Frequency Distribution Table

Interval	Frequency
[900,1000)	1
[1000,1100)	10
[1100,1200)	4
[1200,1300)	3
[1300,1400)	1
[1400,1500)	1

Relative Frequency Distribution Table

Interval	Frequency	Relative Frequency
[900,1000)	1	0.05
[1000,1100)	10	0.5
[1100,1200)	4	0.2
[1200,1300)	3	0.15
[1300,1400)	1	0.05
[1400,1500)	1	0.05

Selected Descriptive Statistics

1) Sample Mean = (1005+963+1035+1027+1281+1272+1057+1079+1034+1070+1173+1079+1067+1104+1347+1439+1029+1100+1204+1160)/20 = 1126.25 cm³
Sample Median = 1079 cm³
Sample Mode = 1079 cm³
Sample midrange = (1439+963/2) = 1201 cm³

2) Range = 476 cm³

Sample variance = (((1005-1126.25)^2)+((963-1126.25)^2)+((1035-1126.25)^2)+((1027-1126.25)^2)+((1281-1126.25)^2)+((1272-1126.25)^2)+((1057-1126.25)^2+((1079-1126.25)^2)+((1034-1126.25)^2)+((1070-1126.25)^2)+((1173-1126.25)^2)+((1079-1126.25)^2)+((1067-1126.25)^2)+((1104-1126.25)^2)+((1347-1126.25)^2)+((1439-1126.25)^2)+((1029-1126.25)^2)+((1100-1126.25)^2)+((1204-1126.25)^2)+((1160-1126.25)^2))/20 = 14785.9875 ≈ 14785.99

Sample standard deviation = (variance)^(1/2) = 121.597646 ≈ 121.60
Sample interquartile range = |(1034+1035)/2 - (1173+1204)/2| = 154

3) Five number summary:

·      Min = 963 cm³

·      First Quartile = 1034.5 cm³

·      Median = 1079 cm³

·      Third Quartile = 1188.5 cm³

·      Max = 1439 cm³

4)         a) 11 values < 1100 cm³

            b) 2 values > 1300 cm³

            c) 1000 cm³ < 17 values < 1300 cm³

Extensions:

2)         a) 11/20 = 0.55 = 55%

            b) 2/20 = 0.1 = 10%

            c) 17/20 = 0.85 = 85%