Problem 35 - Algebra A Combined Function

Question

Minh, a psychology student, kept a record of how much time she spent studying for each of her 20 -point psychology quizzes and her score on each quiz. $$ \begin{array}{|l|c|c|c|c|c|c|c|c|} \hline \text { Hours Spent Studying } & 0.50 & 0.75 & 1.00 & 1.25 & 1.50 & 1.50 & 1.75 & 2.00 \\ \hline \text { Quiz Score } & 10 & 12 & 15 & 16 & 18 & 19 & 19 & 20 \\ \hline \end{array} $$ a. Write the data as ordered pairs of the form (hours spent studying, quiz score). b. In your own words, write the meaning of the ordered pair (1.25,16) C. Create a scatter diagram of the paired data. \(\mathrm{Be}\) sure to label the axes appropriately. d. What might Minh conclude from the scatter diagram?

Step-by-Step Solution

Answer

Minh can conclude that more study hours are associated with higher quiz scores.

Step 1: Writing Ordered Pairs

To write the data as ordered pairs, we pair each hour value with the corresponding quiz score. The ordered pairs from the table are: - (0.50, 10) - (0.75, 12) - (1.00, 15) - (1.25, 16) - (1.50, 18) - (1.50, 19) - (1.75, 19) - (2.00, 20)

Step 2: Explain an Ordered Pair

The ordered pair (1.25, 16) means that Minh spent 1.25 hours studying for a psychology quiz and scored 16 out of 20 on that quiz.

Step 3: Creating a Scatter Diagram

Plot each ordered pair on a graph where the x-axis represents the 'Hours Spent Studying' and the y-axis represents 'Quiz Score'. Place a point for each ordered pair. Make sure each axis is appropriately labeled.

Step 4: Conclusion from the Scatter Diagram

From the scatter diagram, we can observe a positive correlation: as the hours spent studying increase, the quiz scores also tend to increase. Minh might conclude that more study time is associated with higher quiz scores.

Key Concepts

Ordered Pairs

Ordered pairs are a fundamental concept in mathematics, particularly useful in graphing and representing data. They consist of two elements, which we can denote generally as

• (x, y)

The first element, x, is often the independent variable, and the second, y, is usually the dependent variable.
This notation helps us understand the relationship between two quantities. In the context of Minh's study record:

• (Hours Spent Studying, Quiz Score)

Each pair directly shows how much time she spent studying and the corresponding score she achieved on her quiz. These pairs can be visually represented on a graph to better understand potential relationships. Writing data as ordered pairs simplifies how we analyze and present it.

Positive Correlation

A positive correlation occurs when two variables move in the same direction. In simple terms, as one variable increases, the other variable also increases.
This is easily observable in Minh's study hours and quiz scores. As she spends more hours studying, her quiz scores tend to improve. In mathematical terms, this indicates a positive relationship between study time and quiz performance.
Recognizing positive correlation is useful in predicting and interpreting data. When viewing a graph such as a scatter plot:

• Points should form an upwards trend from left to right.
• This suggests that increasing one variable, like study hours, typically results in increasing the other variable, like quiz scores.

Interpreting Data

Interpreting data involves analyzing information to extract meaning or insights. In Minh's situation, by observing the ordered pairs and the resulting graph, we can see a clear pattern: more study time correlates with higher quiz scores.
This interpretation suggests a causal relationship, encouraging the idea that putting in more study hours may enhance performance. With a scatter diagram, interpretation is enriched because:

• One can visually connect data points to observate trends.
• This aids in making predictions based on visible patterns.

Furthermore, interpreting data correctly allows us to make informed decisions, like adjusting study habits to improve grades.

Hours vs. Quiz Scores

The comparison of hours spent studying versus the quiz scores is at the heart of understanding Minh's performance. This relationship is valuable when aiming to optimize study habits for better results.
By plotting the data on a scatter plot with 'Hours Spent Studying' on the x-axis and 'Quiz Score' on the y-axis, we gain a visual representation of how these two variables interact. The plotted points show us:

• As hours increase from 0.50 to 2.00, scores tend to rise from 10 to 20.
• This indicates that more study time aligns with improved quiz scores.

Understanding this link can benefit students by demonstrating the tangible benefits of dedicating more time to studying. It encourages effective time management for better academic performance.