91Ó°ÊÓ

The U.S. Open Golf Tournament was played at Congressional Country Club, Bethesda, Maryland, with prizes ranging from \(\$ 465,000\) for first place to \(\$ 5000\). Par for the course was \(70 .\) The tournament consisted of four rounds played on different days. The scores for each round of the 32 players who placed in the money (more than \(\$ 17,000\) ) were given on a web site. For more information, visit the Brase/Brase statistics site at http://www.cengage.com/statistics/brase and find the link to golf. The scores for the first round were as follows: \(\begin{array}{lllllllllll}71 & 65 & 67 & 73 & 74 & 73 & 71 & 71 & 74 & 73 & 71 \\ 70 & 75 & 71 & 72 & 71 & 75 & 75 & 71 & 71 & 74 & 75 \\ 66 & 75 & 75 & 75 & 71 & 72 & 72 & 73 & 71 & 67 & \end{array}\) The scores for the fourth round for these players were as follows: \(\begin{array}{lllllllllll}69 & 69 & 73 & 74 & 72 & 72 & 70 & 71 & 71 & 70 & 72 \\ 73 & 73 & 72 & 71 & 71 & 71 & 69 & 70 & 71 & 72 & 73 \\ 74 & 72 & 71 & 68 & 69 & 70 & 69 & 71 & 73 & 74 & \end{array}\) (a) Make a stem-and-leaf display for the first-round scores. Use two lines per stem. (See Problem 5.) (b) Make a stem-and-leaf display for the fourth-round scores. Use two lines per stem. (c) Compare the two distributions. How do the highest scores compare? How do the lowest scores compare?

Step by step solution

01

Organize First-Round Scores

To create a stem-and-leaf plot for the first round scores, we first organize the scores in ascending order: 65, 66, 67, 67, 70, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71, 72, 72, 72, 73, 73, 73, 73, 74, 74, 74, 75, 75, 75, 75, 75, 75, 75.

02

Construct Stem-and-Leaf Plot for First Round

For the first round, use two lines per stem: - Stems: Based on tens digits (6 and 7) - Leaves: Based on units digits ``` 6 | 5 6 | 6, 7, 7 7 | 0, 1, 1, 1, 1, 1, 1, 1, 1 7 | 2, 2, 2 7 | 3, 3, 3, 3 7 | 4, 4, 4 7 | 5, 5, 5, 5, 5, 5, 5 ```

03

Organize Fourth-Round Scores

To create a stem-and-leaf plot for the fourth-round scores, we first organize the scores in ascending order: 68, 69, 69, 69, 69, 70, 70, 70, 71, 71, 71, 71, 71, 71, 71, 71, 72, 72, 72, 72, 72, 73, 73, 73, 73, 74, 74, 74.

04

Construct Stem-and-Leaf Plot for Fourth Round

For the fourth round, use two lines per stem: - Stems: Based on tens digits (6 and 7) - Leaves: Based on units digits ``` 6 | 8, 9, 9, 9, 9 7 | 0, 0, 0 7 | 1, 1, 1, 1, 1, 1, 1, 1 7 | 2, 2, 2, 2, 2 7 | 3, 3, 3, 3 7 | 4, 4, 4 ```

05

Compare Distributions

Comparing both plots: - Highest scores: First round has more scores in the 74-75 range compared to the fourth round. - Lowest scores: Fourth round has slightly lower scores with scores of 68 and 69, while the first round's lowest score is 65, but with more scores around 70-71.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Statistical Comparison

When tackling a comparison between two sets of data, like the golf scores from the first and fourth rounds of a tournament, statistical comparison plays a vital role. It allows us to discern patterns and differences that would otherwise be difficult to spot.
By using simple tools such as stem-and-leaf plots, we can easily make these comparisons. Stem-and-leaf plots organize data, making it easy to see where scores are concentrated.
In comparing the highest scores from both rounds, we notice that more players scored in the higher 74-75 range in the first round, hinting that conditions or player performance might have been tougher in the fourth round.

• Highest first-round scores were 74 and 75
• Highest fourth-round scores were around 73 and 74

These subtle numerical differences indicate varying levels of difficulty or player fatigue and can have profound implications in understanding how scores fluctuate over the progression of a tournament.

Data Organization

Data organization is a fundamental skill in statistics that helps in transforming raw data into meaningful information. It involves arranging data so that it can be analyzed effectively. In exercises like organizing golf scores, using a stem-and-leaf plot is beneficial.
The stem-and-leaf plot is especially useful because it retains the original data while also giving a quick view of its distribution.

• Stems represent the tens digit of scores.
• Leaves represent the units digit.

For instance, in organizing the first round scores, we see several players tightly grouped around the par of 70 with multiple scores of 71, which highlights the common performance level.
In contrast, for the fourth round, grouping showed more diversity with scores stretching down to 68, indicating a shift in player performance or conditions.

Distribution Analysis

Analyzing the distribution of data is crucial in understanding how data behaves over a set of conditions, like rounds in a golf tournament. Distribution analysis involves looking at how frequently each score occurs, which gives insight into central tendencies and variabilities.
In the given golf scores scenario, distribution analysis through stem-and-leaf plots shows us that:

• First round had a concentration of scores at 71, showing it as a common score.
• Fourth round showed more spread, with a wider range of lower-end scores.

The analysis also reveals scores' spread and central tendency. For example, in the first round, scores like 74 and 75 were more common compared to the fourth round where low scores like 68 were more frequent.
Understanding this distribution helps in exploring factors that might influence players' performances, like weather conditions or psychological pressure during the final rounds of a tournament.