College

The percentage of cotton in material used to manufacture men's shirts is as follows:

33.2, 37.8, 34.6, 33.0, 32.8, 36.1, 34.3, 35.5, 32.9, 36.6, 34.1, 33.0, 33.2, 36.4, 32.5, 34.7, 34.7, 37.6, 33.6, 33.5, 33.9, 37.4, 32.8, 34.0, 35.1, 32.1, 35.6, 37.4, 35.1, 35.0, 32.1, 32.8, 36.7, 34.6, 37.1, 34.6, 33.8, 37.0, 34.5, 35.5, 33.2, 36.3, 36.4, 32.5, 35.2, 32.0, 35.5, 36.7, 34.1, 33.8, 36.5, 32.8, 35.5, 32.4, 36.4, 34.0, 32.2, 35.6, 32.5, 34.4, 32.9, 35.0, 35.4, 33.2

Tasks:
a. Construct a stem-and-leaf display for the data.
b. Calculate the median and quartiles of these data.

Answer :

The answer provides a stem-and-leaf display for the data and calculates the median and quartiles. It offers a clear step-by-step explanation to help understand the concepts.

a. Stem-and-leaf display:

32 | 0 1 1 2 2 4 4 5 5 5 8 8 8 8 9 9

33 | 0 0 2 2 2 5 5 6 6 6 8 8 8

34 | 0 0 0 1 1 3 3 4 4 5 5 5 6 6 6 6 7 7 7 7 9

35 | 0 0 1 1 1 2 2 4 5 5 5 5 6

36 | 1 3 4 4 5 5 6 7 7

37 | 0 0 1 4 4 6

b. Median and Quartiles:

Sorted data: 32.0, 32.1, 32.1, 32.2, ..., 37.6, 37.8, 37.8, 67 values

Median: (67+1)/2 = 34th value = 34.1%

Q1: (25+1)/4 = 17th value = 33.8%

Q3: 3(25+1)/4 = 51st value = 35.5%

Answer:

Median = 34.55

Quartiles:

Q1 = 33.1

Q2 = 34.55

Q3 = 35.6

Inter Quartile range = 2.5

Step-by-step explanation:

Step 1

We arrange the given the following data from lowest to the highest

32.0, 32.1, 32.1, 32.2, 32.4, 32.5, 32.5, 32.5, 32.8, 32.8, 32.8, 32.8, 32.9, 32.9, 33.0, 33.0, 33.2, 33.2, 33.2, 33.2, 33.5, 33.6, 33.8, 33.8, 33.9, 34.0, 34.0, 34.1, 34.1, 34.3, 34.4, 34.5, 34.6, 34.6, 34.6, 34.7, 34.7, 35.0, 35.0, 35.1, 35.1, 35.2, 35.4, 35.5, 35.5, 35.5, 35.5, 35.6, 35.6, 36.1, 36.3, 36.4, 36.4, 36.4, 36.5, 36.6, 36.7, 36.7, 37.0, 37.1, 37.4, 37.4, 37.6, 37.8

a. Construct a stem-and-leaf display for the data.

Stem and leaf Display

Stem | Leaf

32 | 0,1,1,2,4,5,5,5,8,8,8,8,9,9

33 | 0,0,2,2,2,2,5,6,8,8,9

34 | 0,0,1,1,3,4,5,6,6,6,7,7

35 | 0,0,1,1,2,4,5,5,5,5,6,6,

36 | 1,3,4,4,4,5,6,7,7,

37 | 0,1,4,4,6,8

b. Calculate the median and quartiles of these data.

Number of terms = 64

1) Median = 1/2(n + 1)th value

n = 64

= 1/2(64 + 1)th

= 1/2(65)th

= 32.5 th value

This means it is between the 32nd and 33rd value

32nd = 34.5

33rd = 34.6

= 34.5 + 34.6/2

= 69.1/2

= 34.55

2) First Quartile

1/4(n + 1)th value

n = 64

= 1/4(64 + 1)th

= 1/4(65)th

= 16.5th value

This means it is between the 16th and 17th value

16th value = 33.0

17tj value = 33.2

= 33.0 + 33.2/2

= 66.2/2

= 34.55

Q1 --> 33.1

3)Second Quartile = Median

1/2(n + 1)th value

n = 64

= 1/2(64 + 1)th

= 1/2(65)th

= 32.5 th value

This means it is between the 32nd and 33rd value

32nd = 34.5

33rd = 34.6

= 34.5 + 34.6/2

= 69.1/2

= 34.55

Q2 --> 34.55

Third Quartile

3/4(n + 1)th value

n = 64

= 3/4(64 + 1)th

= 3/4(65)th

= 48.75 th value

This means it is towards the 49th value

32nd = 34.5

Hence,

Q3 --> 35.6

Inter Quartile range

Q3 - Q1

= 35.6 - 33.1

= 2.5