Chapter 2.2, Problem 31E

Hail to the chief: There have been 58 presidential inaugurations in U.S. history. At each one, the president has made an inaugural address. Following are the number of words spoken in each of these addresses.

1. Construct a frequency distribution with approximately five classes.
2. Construct a frequency histogram from the frequency distribution in part (a).
3. Construct a relative frequency distribution using the same classes as in part (a).
4. Construct a relative frequency histogram from this relative frequency distribution.
5. Are the histograms skewed to the left, skewed to the right, or approximately symmetric?
6. Construct a frequency distribution with approximately nine classes.
7. Repeat parts (b)—(d), using the frequency distribution constructed in part (f).
8. Do you think that five and nine classes are both reasonably good choices for these data, or do you think that one choice is much better than the other? Explain your reasoning.

To determine

To construct:A frequency distribution with approximately 5 classes.

Explanation of Solution

Given information:There have been 58 presidential inaugurations in U.S. history. At each one, the president has made an inaugural address. Following are the number of words spoken in each of these addresses.

1431	135	2321	1730	2166	1177	1211	3375
4472	2915	1128	1176	3843	8460	4809	1090
3336	2831	3637	700	1127	1339	2486	2979
1686	4392	2015	3968	2218	984	5434	1704
1526	3329	4055	3672	1880	1808	1359	559
2273	2459	1658	1366	1507	2128	1803	1229
2427	2561	2320	1598	2155	1592	2071	2395
2096	1433

Definition used: Frequency distributions for quantitative data are just like those for qualitative data, except the data are divided into classes rather categories.

Solution:

The class width is 2000. The minimum and maximum values of the ratings are 0 and 9999.

The table of frequency distribution is given by

Number of words	Frequency
0-1999	27
2000-3999	25
4000-5999	5
6000-7999	0
8000-9999	1

To determine

To construct:A frequency histogram from the frequency distribution.

Explanation of Solution

Given information:The table of frequency distribution is given by

Number of words	Frequency
0-1999	27
2000-3999	25
4000-5999	5
6000-7999	0
8000-9999	1

Definition used: Histograms based on frequency distributions are called frequency histogram.

Solution:

The frequency histogram for the given data is given by

  

To determine

To construct: A relative frequency distribution.

Explanation of Solution

Given information:The table of frequency distribution is given by

Number of words	Frequency
0-1999	27
2000-3999	25
4000-5999	5
6000-7999	0
8000-9999	1

Formula used: $Relativefrequency=\frac{Frequency}{ Sum\text{ of all }Frequency}$

Solution:

From the given table,

The sum of all frequency is $27+25+5+0+1=58$

The table of relative frequency is given by

Number of words	Frequency	Relative frequency
0-1999	27	$\frac{27}{58}=0.466$
2000-3999	25	$\frac{25}{58}=0.431$
4000-5999	5	$\frac{5}{58}=0.086$
6000-7999	0	$\frac{0}{58}=0.000$
8000-9999	1	$\frac{1}{58}=0.017$

To determine

To construct: A relative frequency histogram.

Explanation of Solution

Given information:There have been 58 presidential inaugurations in U.S. history. At each one, the president has made an inaugural address. Following are the number of words spoken in each of these addresses.

1431	135	2321	1730	2166	1177	1211	3375
4472	2915	1128	1176	3843	8460	4809	1090
3336	2831	3637	700	1127	1339	2486	2979
1686	4392	2015	3968	2218	984	5434	1704
1526	3329	4055	3672	1880	1808	1359	559
2273	2459	1658	1366	1507	2128	1803	1229
2427	2561	2320	1598	2155	1592	2071	2395
2096	1433

Definition used: Histograms based on relative frequency distributions are called relative frequency histogram.

Solution:

Number of words	Relative Frequency
0-1999	0.466
2000-3999	0.431
4000-5999	0.086
6000-7999	0.000
8000-9999	0.017

Therelative frequency histogram for the given data is given by

  

To determine

To find: Whether the histograms are skewed to the right, skewed to the left, or approximately symmetric.

Answer to Problem 31E

The histogram is skewed to the right.

Explanation of Solution

Given information:There have been 58 presidential inaugurations in U.S. history. At each one, the president has made an inaugural address. Following are the number of words spoken in each of these addresses.

1431	135	2321	1730	2166	1177	1211	3375
4472	2915	1128	1176	3843	8460	4809	1090
3336	2831	3637	700	1127	1339	2486	2979
1686	4392	2015	3968	2218	984	5434	1704
1526	3329	4055	3672	1880	1808	1359	559
2273	2459	1658	1366	1507	2128	1803	1229
2427	2561	2320	1598	2155	1592	2071	2395
2096	1433

Definition used:

A histogram which has a long right-hand tail is said to be skewed to the right.

A histogram which has a long left-hand tail is said to be skewed to the left.

A histogram is symmetric if its right half is a minor image of its left half.

Solution:

The frequency histogram for the given data is given by

  

The above histogram has a right-hand tail; therefore, it is skewed to the right.

Hence, the histogram is skewed to the right.

To determine

To construct: A frequency distribution with approximately 9 classes.

Explanation of Solution

Given information: There have been 58 presidential inaugurations in U.S. history. At each one, the president has made an inaugural address. Following are the number of words spoken in each of these addresses.

1431	135	2321	1730	2166	1177	1211	3375
4472	2915	1128	1176	3843	8460	4809	1090
3336	2831	3637	700	1127	1339	2486	2979
1686	4392	2015	3968	2218	984	5434	1704
1526	3329	4055	3672	1880	1808	1359	559
2273	2459	1658	1366	1507	2128	1803	1229
2427	2561	2320	1598	2155	1592	2071	2395
2096	1433

Definition used: Frequency distributions for quantitative data are just like those for qualitative data, except the data are divided into classes rather categories.

Solution:

Let us take the class width is 1000. The minimum and maximum values of the ratings are 0 and 8999.

The table of frequency distribution is given by

Number of words	Frequency
0-999	4
1000-1999	23
2000-2999	18
3000-3999	7
4000-4999	4
5000-5999	1
6000-6999	0
7000-7999	0
8000-8999	1

To determine

To construct: A frequency histogram, relative frequency distribution and relative frequency histogram.

Explanation of Solution

Given information: The table of frequency distribution is given by

Number of words	Frequency
0-999	4
1000-1999	23
2000-2999	18
3000-3999	7
4000-4999	4
5000-5999	1
6000-6999	0
7000-7999	0
8000-8999	1

Solution:

The frequency histogram for the given data is given by

  

The sum of all frequency is 58

The relative frequency distribution table is given by

Number of words	Frequency	Relative frequency
0-999	4	$\frac{4}{58}=0.069$
1000-1999	23	$\frac{23}{58}=0.397$
2000-2999	18	$\frac{18}{58}=0.310$
3000-3999	7	$\frac{7}{58}=0.121$
4000-4999	4	$\frac{4}{58}=0.069$
5000-5999	1	$\frac{1}{58}=0.017$
6000-6999	0	$\frac{0}{58}=0.000$
7000-7999	0	$\frac{0}{58}=0.000$
8000-8999	1	$\frac{1}{58}=0.017$

The relative frequency histogram for the given data is given by

  

To determine

To explain: Whether the good choices for the data are that classes with 5 or 9.

Answer to Problem 31E

The one with 9 classes is the good choice than 5 classes

Explanation of Solution

Given information:There have been 58 presidential inaugurations in U.S. history. At each one, the president has made an inaugural address. Following are the number of words spoken in each of these addresses.

1431	135	2321	1730	2166	1177	1211	3375
4472	2915	1128	1176	3843	8460	4809	1090
3336	2831	3637	700	1127	1339	2486	2979
1686	4392	2015	3968	2218	984	5434	1704
1526	3329	4055	3672	1880	1808	1359	559
2273	2459	1658	1366	1507	2128	1803	1229
2427	2561	2320	1598	2155	1592	2071	2395
2096	1433

The one with 9 classes is more appropriate than the one with only 5 classes.

This is because the one with only 5 classes is too wide and only the most basic features of the data are visible.

Hence, the one with 9 classes is the good choice than 5 classes.