Question

2. A. Find the five-number summary for the Tar content of the sample of cigarettes. List the values. Draw a boxplot to illustrate this. Be sure to show an evenly marked scale below it. Use a ruler to draw the figure. Comment on its shape, i.e. does it appear to be skewed left, skewed right, or symmetrical? B. Compute the IQR, UF, and LF for the Tar content of the sample of cigarettes. Does the data have any outliers? If so, list them. C. State the mode(s) of the tar content. D. Compute the range of the tar content.

Cigarette Data (all quantities are in mg per cigarette)

Brand	Nicotine	Tar	Carbon Monoxide
American Filter	16	1.2	15
Benson and Hedges	16	1.2	15
Camel	16	1.0	17
Capri	9	0.8	6
Carlton	1	0.1	1
Cartier Vendome	8	0.8	8
Chelsea	10	0.8	10
GPC Approved	16	1.0	17
Hi-Lite	14	1.0	13
Kent	13	2.0	13
Lucky Strike	13	1.1	13
Malibu	15	1.2	15
Marlboro	16	1.2	15
Merit	9	0.7	11
Newport Stripe	11	0.9	15
Now	2	0.2	3
Old Gold	18	1.4	18
Pall Mall	15	1.2	15
Players	13	1.1	12
Raleigh	15	1.0	16
Richland	17	1.3	16
Rite	9	0.8	10
Silva Thins	12	1.0	10
Tareyton	14	1.0	17
Triumph	5	0.5	7
True	6	0.6	7
Vantage	8	0.7	11
Viceroy	18	1.4	15
Winston	16	1.1	18

Answer 1

First we order the sample on Tar content in the ascending order as gien below

Brand	Nicotine	Tar	Carbon Monoxide
Carlton	1	0.1	1
Now	2	0.2	3
Triumph	5	0.5	7
TRUE	6	0.6	7
Merit	9	0.7	11
Vantage	8	0.7	11
Capri	9	0.8	6
Cartier Vendome	8	0.8	8
Chelsea	10	0.8	10
Rite	9	0.8	10
Newport Stripe	11	0.9	15
Camel	16	1	17
GPC Approved	16	1	17
Hi-Lite	14	1	13
Raleigh	15	1	16
Silva Thins	12	1	10
Tareyton	14	1	17
Lucky Strike	13	1.1	13
Players	13	1.1	12
Winston	16	1.1	18
American Filter	16	1.2	15
Benson and Hedges	16	1.2	15
Malibu	15	1.2	15
Marlboro	16	1.2	15
Pall Mall	15	1.2	15
Richland	17	1.3	16
Old Gold	18	1.4	18
Viceroy	18	1.4	15
Kent	13	2	13

The value of Tar in the first row is 0.1 and is the minimum value of Tar

The Tar value for the last row is 2 and is the maximum value of Tar

There are n=29 observations. When n is odd, (n+1)/2 th observation is the median. That is (29+1)/2 = 15th observation in the median. Median = 1

The first quartile Q1 is the observation on 25th percentile.. There are 15 observations which are less than the median, including the mediamn. Q1 is the median of this lower protion. The median of the lower portion is (15+1)/2 = 16/2 = 8th observation. So Q1= 0.8

The third quartile Q3 is the observation on 75th percentile.. There are 15 observations which are more than the median, including the mediamn. Q3 is the median of this upper protion. The median of the upper portion is (15+1)/2 = 16/2 = 8th observation starting from median which is the 15th observation. That means Q3 is the 22nd observation. So Q3=1.2

The 5 number summary is

Minimum	0.1
First quartile Q1	0.8
Median	1
3rd quartile Q3	1.2
Maximum	2

B) Inter Quartile Range IQR is

The lower fence LF is

Upper fence UF is

The observations which are lower than LF and above UF can be considered as outliers. We have Carlton with Tar = 0.1 and Kent with Tar = 2 as outliers

The box plot can be plotted as

From the plot we can see that the median is exactly at the center of the box. Median is at 1, and Q1 is 0.2 towards the left at 0.8 and Q3 is 0.2 towards the right at 1.2. That means the distribution of Tar is symmetrical about the median

C) Mode is the value with highest number of observations against it.

Tar content	Frequency
0.1	1
0.2	1
0.5	1
0.6	1
0.7	2
0.8	4
0.9	1
1	6
1.1	3
1.2	4
1.3	1
1.4	2
2	1

We can see that Tar = 1 has 6 observations against it. That means mode = 1

D)The range is maximum - minimum = 2-0.1 = 1.9