Chapter 2.2, Problem 26E

Batting average: The following frequency distribution presents the batting averages of Major League Baseball players in both the American League and the National League who had 300 or more plate appearances during a recent season.

1. Construct a frequency histogram for the American League.
2. Construct a frequency histogram for the National League.
3. Construct a relative frequency distribution for the American League.
4. Construct a relative frequency distribution for the National League.
5. Construct a relative frequency histogram for the American League.
6. Construct a relative frequency histogram for the National League.
7. What percentage of American League players had batting averages of 0.300 or more?
8. What percentage of National League players had batting averages of 0.300 or more?
9. Compare the relative frequency histograms. What is the main difference between the distributions of batting averages in the two leagues?

To determine

To construct: A frequency histogram for American League.

Explanation of Solution

Given information:The following frequency distribution presents the batting averages of Major League Baseball players in both the American League and the National League who had 300 or more plate appearances during a recent season.

Batting average	American LeagueFrequency	National LeagueFrequency
0.180-0.199	2	2
0.200-0.219	7	7
0.220-0.239	21	20
0.240-0.259	30	29
0.260-0.279	26	32
0.280-0.299	21	30
0.300-0.319	12	17
0.320-0.339	5	5
0.340-0.359	0	1

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

Solution:

The following frequency distribution presents the batting averages of Major League Baseball players in the American Leaguewho had 300 or more plate appearances during a recent season.

Batting average	American LeagueFrequency
0.180-0.199	2
0.200-0.219	7
0.220-0.239	21
0.240-0.259	30
0.260-0.279	26
0.280-0.299	21
0.300-0.319	12
0.320-0.339	5
0.340-0.359	0

The frequency histogram for American League is given by

  

To determine

To construct: A frequency histogram for National League.

Explanation of Solution

Given information:The following frequency distribution presents the batting averages of Major League Baseball players in both the American League and the National League who had 300 or more plate appearances during a recent season.

Batting average	American LeagueFrequency	National LeagueFrequency
0.180-0.199	2	2
0.200-0.219	7	7
0.220-0.239	21	20
0.240-0.259	30	29
0.260-0.279	26	32
0.280-0.299	21	30
0.300-0.319	12	17
0.320-0.339	5	5
0.340-0.359	0	1

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

Solution:

The following frequency distribution presents the batting averages of Major League Baseball players in the American Leaguewho had 300 or more plate appearances during a recent season.

Batting average	National LeagueFrequency
0.180-0.199	2
0.200-0.219	7
0.220-0.239	20
0.240-0.259	29
0.260-0.279	32
0.280-0.299	30
0.300-0.319	17
0.320-0.339	5
0.340-0.359	1

The frequency histogram for American League is given by

  

To determine

To construct: A relative frequency distribution for American League.

Explanation of Solution

Given information:The following frequency distribution presents the batting averages of Major League Baseball players in both the American League and the National League who had 300 or more plate appearances during a recent season.

Batting average	American LeagueFrequency	National LeagueFrequency
0.180-0.199	2	2
0.200-0.219	7	7
0.220-0.239	21	20
0.240-0.259	30	29
0.260-0.279	26	32
0.280-0.299	21	30
0.300-0.319	12	17
0.320-0.339	5	5
0.340-0.359	0	1

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

Solution:

From the given table,

The sum of all frequency for American League is $2+7+21+30+26+21+12+5+0=124$

The table of relative frequency is given by

Batting average	American LeagueFrequency	American LeagueRelative frequency
0.180-0.199	2	$\frac{2}{124}=0.016$
0.200-0.219	7	$\frac{7}{124}=0.056$
0.220-0.239	21	$\frac{21}{124}=0.169$
0.240-0.259	30	$\frac{30}{124}=0.242$
0.260-0.279	26	$\frac{26}{124}=0.210$
0.280-0.299	21	$\frac{21}{124}=0.169$
0.300-0.319	12	$\frac{12}{124}=0.097$
0.320-0.339	5	$\frac{5}{124}=0.040$
0.340-0.359	0	$\frac{0}{124}=0.000$

The relative frequency for the American League is given by

Batting average	American LeagueRelative frequency
0.180-0.199	0.016
0.200-0.219	0.056
0.220-0.239	0.169
0.240-0.259	0.242
0.260-0.279	0.210
0.280-0.299	0.169
0.300-0.319	0.097
0.320-0.339	0.040
0.340-0.359	0.000

To determine

To construct: A relative frequency distribution for National League.

Explanation of Solution

Given information:The following frequency distribution presents the batting averages of Major League Baseball players in both the American League and the National League who had 300 or more plate appearances during a recent season.

Batting average	American LeagueFrequency	National LeagueFrequency
0.180-0.199	2	2
0.200-0.219	7	7
0.220-0.239	21	20
0.240-0.259	30	29
0.260-0.279	26	32
0.280-0.299	21	30
0.300-0.319	12	17
0.320-0.339	5	5
0.340-0.359	0	1

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

Solution:

From the given table,

The sum of all frequency for National League is $2+7+20+29+32+30+17+5+1=143$

The table of relative frequency is given by

Batting average	National LeagueFrequency	National LeagueRelative frequency
0.180-0.199	2	$\frac{2}{143}=0.014$
0.200-0.219	7	$\frac{7}{143}=0.049$
0.220-0.239	20	$\frac{20}{143}=0.140$
0.240-0.259	29	$\frac{29}{143}=0.203$
0.260-0.279	32	$\frac{32}{143}=0.224$
0.280-0.299	30	$\frac{30}{143}=0.210$
0.300-0.319	17	$\frac{17}{143}=0.119$
0.320-0.339	5	$\frac{5}{143}=0.035$
0.340-0.359	1	$\frac{1}{143}=0.007$

The relative frequency for the NationalLeague is given by

Batting average	National LeagueRelative frequency
0.180-0.199	0.014
0.200-0.219	0.049
0.220-0.239	0.140
0.240-0.259	0.203
0.260-0.279	0.224
0.280-0.299	0.210
0.300-0.319	0.119
0.320-0.339	0.035
0.340-0.359	0.007

To determine

To construct: A relative frequency histogram for American League.

Explanation of Solution

Given information:The following frequency distribution presents the batting averages of Major League Baseball players in both the American League and the National League who had 300 or more plate appearances during a recent season.

Batting average	American LeagueFrequency	National LeagueFrequency
0.180-0.199	2	2
0.200-0.219	7	7
0.220-0.239	21	20
0.240-0.259	30	29
0.260-0.279	26	32
0.280-0.299	21	30
0.300-0.319	12	17
0.320-0.339	5	5
0.340-0.359	0	1

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

Solution:

Batting average	American LeagueRelative frequency
0.180-0.199	0.016
0.200-0.219	0.056
0.220-0.239	0.169
0.240-0.259	0.242
0.260-0.279	0.210
0.280-0.299	0.169
0.300-0.319	0.097
0.320-0.339	0.040
0.340-0.359	0.000

The relative frequency histogram for the given data is given by

  

To determine

To construct: A relative frequency histogram for National League.

Explanation of Solution

Given information:The following frequency distribution presents the batting averages of Major League Baseball players in both the American League and the National League who had 300 or more plate appearances during a recent season.

Batting average	American LeagueFrequency	National LeagueFrequency
0.180-0.199	2	2
0.200-0.219	7	7
0.220-0.239	21	20
0.240-0.259	30	29
0.260-0.279	26	32
0.280-0.299	21	30
0.300-0.319	12	17
0.320-0.339	5	5
0.340-0.359	0	1

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

Solution:

Batting average	NationalLeagueRelative frequency
0.180-0.199	0.014
0.200-0.219	0.049
0.220-0.239	0.140
0.240-0.259	0.203
0.260-0.279	0.224
0.280-0.299	0.210
0.300-0.319	0.119
0.320-0.339	0.035
0.340-0.359	0.007

The relative frequency histogram for the given data is given by

  

To determine

To find: The percentage of American League players who had batting averages of 0.300 or more.

Answer to Problem 26E

The percentage of American League players who had batting averages of 0.300 or more is 13.7%.

Explanation of Solution

Given information:The following frequency distribution presents the batting averages of Major League Baseball players in both the American League and the National League who had 300 or more plate appearances during a recent season.

Batting average	American LeagueFrequency	National LeagueFrequency
0.180-0.199	2	2
0.200-0.219	7	7
0.220-0.239	21	20
0.240-0.259	30	29
0.260-0.279	26	32
0.280-0.299	21	30
0.300-0.319	12	17
0.320-0.339	5	5
0.340-0.359	0	1

Calculation:

The relative frequency for American League is given by

Batting average	American LeagueRelative frequency
0.180-0.199	0.016
0.200-0.219	0.056
0.220-0.239	0.169
0.240-0.259	0.242
0.260-0.279	0.210
0.280-0.299	0.169
0.300-0.319	0.097
0.320-0.339	0.040
0.340-0.359	0.000

From the above data, the relative frequencies of batting averages of 0.300 or more are 0.097, 0.040 and 0.000

The sum of all the above relative frequencies is $0.097+0.040+0.000=0.137$ .

Then, its percentage is 13.7%

Hence, the percentage of American League players who had batting averages of 0.300 or more is 13.7%.

To determine

To find: The percentage of National League players who had batting averages of 0.300 or more.

Answer to Problem 26E

The percentage of National League players who had batting averages of 0.300 or more is 16.1%.

Explanation of Solution

Given information:The following frequency distribution presents the batting averages of Major League Baseball players in both the American League and the National League who had 300 or more plate appearances during a recent season.

Batting average	American LeagueFrequency	National LeagueFrequency
0.180-0.199	2	2
0.200-0.219	7	7
0.220-0.239	21	20
0.240-0.259	30	29
0.260-0.279	26	32
0.280-0.299	21	30
0.300-0.319	12	17
0.320-0.339	5	5
0.340-0.359	0	1

Calculation:

The relative frequency for National League is given by

Batting average	NationalLeagueRelative frequency
0.180-0.199	0.014
0.200-0.219	0.049
0.220-0.239	0.140
0.240-0.259	0.203
0.260-0.279	0.224
0.280-0.299	0.210
0.300-0.319	0.119
0.320-0.339	0.035
0.340-0.359	0.007

From the above data, the relative frequencies of batting averages of 0.300 or more are 0.119, 0.035 and 0.007.

The sum of all the above relative frequencies is $0.119+0.035+0.007=0.161$ .

Then, its percentage is 16.1%

Hence, the percentage of National League players who had batting averages of 0.300 or more is 16.1%.

To determine

To find: The percentage of players who had batting averages less than 0.220.

Answer to Problem 26E

The batting averages tend to be a bit higher in the National League.

Explanation of Solution

Given information:The following frequency distribution presents the batting averages of Major League Baseball players in both the American League and the National League who had 300 or more plate appearances during a recent season.

Batting average	American LeagueFrequency	National LeagueFrequency
0.180-0.199	2	7
0.200-0.219	7	7
0.220-0.239	21	20
0.240-0.259	30	29
0.260-0.279	26	32
0.280-0.299	21	30
0.300-0.319	12	17
0.320-0.339	5	5
0.340-0.359	0	1

Solution:

The relative frequency histogram for American League is given by

  

The relative frequency histogram for the National League is given by

  

From the above two histograms, we can see that the relative frequency for National league is bit higher than the National League.

Hence, the batting averages tend to be a bit higher in the National League.