Over the past 6 ​seasons, one baseball​ player's batting averages were 0.202​, 0.364​, 0.333​, 0.276​, 0.281​, and 0.277. A second​ player's batting averages were 0.324​, 0.338​, 0.296​, 0.238​, 0.327​, and 0.196. What are the range and mean of each​ player's batting​ averages? Use your results to compare the​ players' batting skills.

Question

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Part 1
Find the range and mean of the first​ player's batting averages.
The range is
  
0.162. ​(Type an integer or a​ decimal.)
The mean is
  
0.289. ​(Round to the nearest thousandth as​ needed.)
Part 2
Find the range and mean of the second​ player's batting averages.
The range is
  
0.142. ​(Type an integer or a​ decimal.)
The mean is
  
0.287. ​(Round to the nearest thousandth as​ needed.)
Part 3
Compare the​ players' batting skills. Choose the correct answer below.
A.
The first player has a greater average batting average during the 6​-year ​period, and the batting averages for the second player are more spread out.
B.
Both players have similar batting averages during the 6​-year ​period, but the batting averages for the second player are more spread out.
C.
Both players have similar batting averages during the 6​-year ​period, but the batting averages for the first player are more spread out.
D.
The second player has a greater average batting average during the 6​-year ​period, and the batting averages for the first player are more spread out.

Answer 1

To determine the range and mean of each player's batting averages, we first need to calculate these values for the two players.

Player 1's Batting Averages:

• Batting Averages: 0.202, 0.364, 0.333, 0.276, 0.281, 0.277

Step 1: Calculate the mean \[ \text{Mean} = \frac{\text{Sum of batting averages}}{\text{Number of seasons}} = \frac{0.202 + 0.364 + 0.333 + 0.276 + 0.281 + 0.277}{6} \] \[ = \frac{1.733}{6} \approx 0.288833 \text{ (round to 0.289)} \]

Step 2: Calculate the range

• Maximum batting average = 0.364
• Minimum batting average = 0.202

\[ \text{Range} = \text{Max} - \text{Min} = 0.364 - 0.202 = 0.162 \]

Part 1 Results:

• The range is: 0.162
• The mean is: 0.289

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Player 2's Batting Averages:

• Batting Averages: 0.324, 0.338, 0.296, 0.238, 0.327, 0.196

Step 1: Calculate the mean \[ \text{Mean} = \frac{0.324 + 0.338 + 0.296 + 0.238 + 0.327 + 0.196}{6} \] \[ = \frac{1.719}{6} \approx 0.2865 \text{ (round to 0.287)} \]

Step 2: Calculate the range

• Maximum batting average = 0.338
• Minimum batting average = 0.196

\[ \text{Range} = \text{Max} - \text{Min} = 0.338 - 0.196 = 0.142 \]

Part 2 Results:

• The range is: 0.142
• The mean is: 0.287

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Part 3: Compare the Players' Batting Skills

• Player 1 Mean: 0.289
• Player 2 Mean: 0.287
• Player 1 Range: 0.162 (spread)
• Player 2 Range: 0.142 (spread)

Comparison:

• The first player has a slightly greater average batting average (0.289 > 0.287).
• The range of the first player’s batting averages (0.162) is greater than that of the second player’s ranges (0.142), indicating that Player 1's averages are more spread out.

Correct Answer:

A. The first player has a greater average batting average during the 6-year period, and the batting averages for the second player are more spread out.