Problem 19 - Algebra 2 and Trigonometry

Question

The grades on a math test of 25 students are listed below. $$\begin{array}{llllllllllll}{86} & {92} & {77} & {84} & {75} & {95} & {66} & {88} & {84} & {53} & {98} & {87} & {83} \\ {74} & {61} & {82} & {93} & {98} & {87} & {77} & {86} & {58} & {72} & {76} & {89}\end{array}$$ a. Organize the data in a stem-and-leaf diagram. b. Organize the data in a frequency distribution table. c. How many students scored 70 or above on the test? d. How many students scored 60 or below on the test?

Step-by-Step Solution

Answer

21 students scored 70 or above; 4 students scored 60 or below.

Step 1: Create a Stem-and-Leaf Diagram

Separate each score into a stem and a leaf. The stem represents the tens digit, while the leaf represents the ones digit. For instance, 86 will be depicted as 8 | 6. Organize all the scores: - **5 |** 3, 8 - **6 |** 1, 6 - **7 |** 2, 4, 5, 6, 7, 7 - **8 |** 2, 3, 4, 4, 6, 6, 7, 7, 8, 9 - **9 |** 2, 3, 5, 8, 8 This diagram provides a visual representation of the data distribution.

Step 2: Construct a Frequency Distribution Table

Group the test scores into ranges, typically of size 10, and count how many scores fall into each range: | Range | Frequency | |-------|-----------| | 50-59 | 2 | | 60-69 | 2 | | 70-79 | 6 | | 80-89 | 10 | | 90-99 | 5 | This table displays how frequently each score range appears in the data set.

Step 3: Count Students Scoring 70 or Above

Refer to the frequency distribution table and add the students who scored 70 or above. The relevant frequency counts are: - 70-79: 6 - 80-89: 10 - 90-99: 5 Total students scoring 70 or above = 6 + 10 + 5 = 21.

Step 4: Count Students Scoring 60 or Below

Refer to the frequency distribution table and add the students who scored 60 or below. The relevant frequency counts are: - 50-59: 2 - 60-69: 2 Total students scoring 60 or below = 2 + 2 = 4.

Key Concepts

Stem-and-Leaf Diagram

A stem-and-leaf diagram is a handy tool often used in mathematics for organizing datasets. It splits numerical data into a "stem," representing the main digits, and "leaf," showing the finer details of each number. For example, in the number 86, the stem would be 8, and the leaf would be 6.

This way of data visualization helps quickly comprehend the shape and distribution of numbers. In the given exercise, by organizing students' test scores into a stem-and-leaf format, we offer an intuitive view of their performance distribution. Each leaf directly follows its stem, and the stems are listed in numerical order.

This structure immediately reveals clusters, gaps, and overall data shape, simplifying the process of identifying key patterns and outliers in the dataset. Feel free to try arranging numbers from other exercises in such diagrams to perfect this skill.

Frequency Distribution Table

A frequency distribution table is another fundamental method in data analysis, designed to present the number of observations for distinct ranges of values. This method simplifies understanding large data sets by displaying them in a concise manner.

For the math test scores, we've organized them into ranges like 50-59, 60-69, etc., showing how many scores fall within each range.

• 50-59: 2 students
• 60-69: 2 students
• 70-79: 6 students
• 80-89: 10 students
• 90-99: 5 students

Such tables give insights not only into the overall data distribution but also help identify where most students scored. This is especially useful in education settings to understand which group of students may need further help or enrichment.

Data Organization

Data organization lies at the heart of effective data analysis. It involves arranging data systematically, making it easier to read, interpret, and derive meaningful conclusions.

In our exercise, data organization played a vital role both in the stem-and-leaf diagram and the frequency distribution table. By properly organizing the scores, we simplify the complex datasets and make them accessible.

Organized data displays clear patterns which might not be visible if the data were presented randomly. This boosts both efficiency and accuracy when analyzing trends, making data organization a core skill to master in any math-related problem-solving situation.

Math Problem-Solving

In the realm of mathematics, problem-solving is the essential process of finding solutions to complex questions. Data analysis tools like stem-and-leaf diagrams and frequency distribution tables aid this by offering clear, organized perspectives on data sets.

For instance, counting students scoring 70 or above required us to analyze the organized data to find 21 students who met that criterion. Similarly, determining how many scored 60 or below was solved in a straightforward manner, showing 4 students in that range.

This problem-solving process underscores the necessity of translating data sets from raw numbers into a structured format that leads to actionable insights. It reflects real-world scenarios where data-driven decisions are pivotal, emphasizing the importance of mastering these techniques to make informed conclusions.