Chapter 2.2, Problem 21E

Quantcast.com provides the number of people from the United States who visit a given website. The list below shows the number of U.S. visitors for the 50 most highly visited websites (Quantcast.com website, December 2015).

Summarize the data by constructing the following:

1. a. A frequency distribution (classes in millions: 20–29.999, 30–30.999, and so on).
2. b. A relative frequency distribution.
3. c. A cumulative distribution.
4. d. A cumulative relative frequency distribution.
5. e. Show a histogram. Comment on the shape of the distribution.
6. f. What is the website with the most U.S. visitors? How many people from the U.S. visited the site?

To determine

Construct a frequency distribution for the data.

Answer to Problem 21E

The frequency distribution is as follows:

Number of visitors in a month	Frequency
20-29.999	18
30-39.999	11
40-49.999	7
50-59.999	2
60-69.999	3
70-79.999	2
80-89.999	2
90-99.999	0
100-199.999	3
200-299.999	2
Total	50

Explanation of Solution

Calculation:

The data represents the number of visitors from country C for the 50 most highly visited websites. The frequency distribution is constructed using the class 20-29.999, 30-39.999, and so on.

Frequency:

The frequencies are calculated using the tally mark and the range of the data is from 0 to 100.999.

• Based on the given information, the class intervals are 20-29.999, 30-39.999, 40-49.999, …, 200-299.999.
• Make a tally mark corresponding to each number of visitors in a month and continue for all values in the data.
• The number of tally marks in each class represents the frequency, f of that class.

Similarly, the frequency of remaining classes for the number of visitors in a month is given below:

Number of visitors in a month	Tally	Frequency
20-29.999	$\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\left|\right||$	18
30-39.999	$\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}|$	11
40-49.999	$\bcancel{\left|\right|\left|\right|}\left|\right|$	7
50-59.999	$\left|\right|$	2
60-69.999	$\left|\right||$	3
70-79.999	$\left|\right|$	2
80-89.999	$\left|\right|$	2
90-99.999		0
100-199.999	$\left|\right||$	3
200-299.999	$\left|\right|$	2
Total		50

To determine

Construct a relative frequency distribution for the data.

Answer to Problem 21E

The relative frequency distribution is as follows:

Number of visitors in a month	Frequency	Relative frequency
20-29.999	18	0.36
30-39.999	11	0.22
40-49.999	7	0.14
50-59.999	2	0.04
60-69.999	3	0.06
70-79.999	2	0.04
80-89.999	2	0.04
90-99.999	0	0.00
100-199.999	3	0.06
200-299.999	2	0.04
Total	50	1.00

Explanation of Solution

Calculation:

Relative frequency:

The general formula for the relative frequency is as follows:

$\text{Relative}\text{Frequency = }\frac{\text{Frequency}}{\text{Total frequency}}$

For the revenue class (20-29.999), substitute frequency as “18” and total frequency as “50”.

$\begin{array}{c}\text{Relative}\text{Frequency}\text{for revenue class(20-29}\text{.999)}=\frac{18}{50}\\ =0.36\end{array}$

Similarly, the relative frequencies for the remaining number of visitors in a month are obtained as follows:

Number of visitors in a month	Frequency	Relative frequency
20-29.999	18	0.36
30-39.999	11	0.22
40-49.999	7	0.14
50-59.999	2	0.04
60-69.999	3	0.06
70-79.999	2	0.04
80-89.999	2	0.04
90-99.999	0	0.00
100-199.999	3	0.06
200-299.999	2	0.04
Total	50	1.00

To determine

Construct a cumulative frequency distribution for the data.

Answer to Problem 21E

The cumulative frequency distribution is as follows:

Number of visitors in a month	Frequency	Cumulative frequency
20-29.999	18	18
30-39.999	11	29
40-49.999	7	36
50-59.999	2	38
60-69.999	3	41
70-79.999	2	43
80-89.999	2	45
90-99.999	0	45
100-199.999	3	48
200-299.999	2	50
Total	50	50

Explanation of Solution

Calculation:

Cumulative frequency:

Cumulative frequency of a particular class is the sum of all frequencies up to that class. The last class’s cumulative frequency is equal to the sample size $n$.

The cumulative frequency for each class is tabulated as follows:

Number of visitors in a month	Relative frequency	Cumulative relative frequency
20-29.999	0.36	0.36
30-39.999	0.22	$0.36+0.22=0.58$
40-49.999	0.14	$0.58+0.14=0.72$
50-59.999	0.04	$0.72+0.04=0.76$
60-69.999	0.06	$0.76+0.06=0.82$
70-79.999	0.04	$0.82+0.04=0.86$
80-89.999	0.04	$0.86+0.04=0.9$
90-99.999	0.00	$0.90+0.00=0.9$
100-199.999	0.06	$0.90+0.06=0.96$
200-299.999	0.04	$0.96+0.04=1.00$
Total	1.00	1.00

To determine

Construct a cumulative relative frequency distribution for the data.

Answer to Problem 21E

The cumulative relative frequency distribution is as follows:

Number of visitors in a month	Cumulative relative frequency
0-49	0.12
50-99	0.70
100-149	0.92
150-199	0.92
200-249	0.94
250-299	0.96
300-349	0.96
350-399	0.96
400-449	1.00

Explanation of Solution

Calculation:

Cumulative relative frequency of a particular class is the sum of all relative frequencies up to that class. The last class’s cumulative relative frequency is equal to the approximate value 1.00.

The relative frequencies for the number of visitors in a month from part (b) are given below:

Number of visitors in a month	Frequency	Relative frequency
20-29.999	18	0.36
30-39.999	11	0.22
40-49.999	7	0.14
50-59.999	2	0.04
60-69.999	3	0.06
70-79.999	2	0.04
80-89.999	2	0.04
90-99.999	0	0.00
100-199.999	3	0.06
200-299.999	2	0.04
Total	50	1.00

The cumulative relative frequencies for the number of visitors in a month are obtained below:

Number of visitors in a month	Relative frequency	Cumulative relative frequency
20-29.999	0.36	0.36
30-39.999	0.22	$0.36+0.22=0.58$
40-49.999	0.14	$0.58+0.14=0.72$
50-59.999	0.04	$0.72+0.04=0.76$
60-69.999	0.06	$0.76+0.06=0.82$
70-79.999	0.04	$0.82+0.04=0.86$
80-89.999	0.04	$0.86+0.04=0.9$
90-99.999	0.00	$0.90+0.00=0.9$
100-199.999	0.06	$0.90+0.06=0.96$
200-299.999	0.04	$0.96+0.04=1.00$
Total	1.00	1.00

To determine

Construct the histogram and comment on the shape of the distribution.

Answer to Problem 21E

• Output using MINITAB software is given below:

The histogram is skewed to the right.

Explanation of Solution

Calculation:

Step by step procedure to draw the frequency histogram chart using MINITAB software is as follows:

• Choose Graph > Histogram.
• Choose Simple, and then click OK.
• In Graph variables, enter the Number of visitors column of data.
• In scale on y-axis, make click on frequency.
• Click on ok.

Skewness:

The data is said to be skewed if there is lack of symmetry, and the values fall on one side that can be either left or right of the distribution.

Right skewed:

If the tail on the distribution is elongated towards the right, and moreover it attains its peak rapidly than its horizontal axis and then it is a right skewed distribution. It is also called positively skewed.

The distribution of number of visitors in the histogram has elongated tail towards right side.

Therefore, the distribution of the histogram with number of visitors is skewed right.

To determine

Find the website with the most number of visitors from country U and find corresponding number.

Answer to Problem 21E

Youtube.com is website with the most number of visitors from country U and the number of people from country U, who visited the site is 222.

Explanation of Solution

From the data set of 50 most highly visited websites, it is observed that highly visited websites is Youtube.com, and number of people from country U that visited the site is 222.

The second highly visited websites is google.com, and number of people from country U that visited the site is 220.