Question

For questions 1-4, use this table of experimental data. $$ \begin{array}{|r|r|r|r|r|r|r|r|r|r|r|} \hline {X} & 2 & 7 & 4 & 5 & 9 & 1 & 6 & 3 & 11 & 8 \\ \hline {Y} & 14 & 47 & 29 & 35 & 63 & 8 & 42 & 22 & 77 & 56 \\ \hline \end{array} $$ What kind of probability model can be made from these data if \(X\) is the input and \(Y\) the result? A. None, the data are random. B. One in which the data are approximately proportional C. One in which the data are inversely proportional D. One in which the data are related by a power of 2

Short answer

B. One in which the data are approximately proportional

Step-by-step solution

- Analyze the Data

First, read the data for both variables, X and Y, from the provided table. The data is as follows: X = {2, 7, 4, 5, 9, 1, 6, 3, 11, 8}, Y = {14, 47, 29, 35, 63, 8, 42, 22, 77, 56}.

- Test for Proportionality

Calculate the ratio \( \frac{Y}{X} \) for each pair of values to see if the results are approximately constant: \( \frac{14}{2} = 7, \frac{47}{7} \approx 6.71, \frac{29}{4} \approx 7.25, \frac{35}{5} = 7, \frac{63}{9} = 7, \frac{8}{1} = 8, \frac{42}{6} = 7, \frac{22}{3} \approx 7.33, \frac{77}{11} = 7, \frac{56}{8} = 7 \).

- Interpret Results of Proportionality Test

The calculated ratios are close to 7, which suggests that Y is approximately proportional to X.

- Check Other Options

Examine the other options. There is no evidence showing that the data is inversely proportional or related by a power of 2. Additionally, the proximity of the ratios to a single value suggests the data are not random.

- Choose the Correct Answer

Since the data indicates proportionality, the correct model is one in which the data are approximately proportional.

Key concepts

Proportionality

Proportionality is a key concept in mathematics that indicates a constant relationship between two variables. When two variables are proportional, their ratio remains constant. For instance, in the provided data set, we have pairs of values for X and Y. To test for proportionality, we calculate the ratio \(\frac{Y}{X}\) for each pair. If these ratios are roughly equal, the variables are considered proportional. In our example: X = {2, 7, 4, 5, 9, 1, 6, 3, 11, 8} and Y = {14, 47, 29, 35, 63, 8, 42, 22, 77, 56}, the ratios calculated were all close to 7, confirming the proportionality between X and Y.

Data Analysis

Data analysis is a process of inspecting, cleaning, transforming, and modeling data to discover useful information and support decision-making. When we analyze the given data, we initially read through and understand both sets of variables, X and Y. For effective data analysis:

• Inspect the data carefully.
• Identify patterns and relationships.
• Test hypotheses, such as checking for proportionality.

In our exercise, we tested proportionality by calculating ratios for each pair, which helped us deduce that Y is approximately proportional to X. This systematic approach aids in drawing accurate conclusions from data.

Probability Models

Probability models are mathematical representations that describe the likelihood of various outcomes. They help in understanding randomness and making predictions. Here we explore the data set: X = {2, 7, 4, 5, 9, 1, 6, 3, 11, 8} and Y = {14, 47, 29, 35, 63, 8, 42, 22, 77, 56}. In evaluating what kind of probability model fits, we calculated the ratios \(\frac{Y}{X}\), finding them close to a constant. This suggests a model in which the data are approximately proportional rather than random, inversely proportional, or related by a power of 2. Therefore, our probability model best describes a proportional relationship between X and Y.

Ratio Calculation

Ratio calculation is crucial in assessing relationships between numbers. A ratio compares the magnitude of two quantities, often revealing their proportionality. In our exercise, we calculated \(\frac{Y}{X}\) for each pair of values. Here's how you do it:

• Select each pair of corresponding values from X and Y.
• Divide Y by X to find the ratio.
• Repeat for all pairs.

For example, \(\frac{14}{2} = 7\) and \(\frac{47}{7} \approx 6.71\). Most of our ratios were around 7, indicating consistency and proportionality. This simple method is powerful in identifying linear relationships in data, like verifying if one variable scales directly with another.