91Ó°ÊÓ

The General Social Survey asks questions about one's happiness and health. One would think that health plays a role in one's happiness. Use the data in the table to determine whether healthier people tend to also be happier. Treat level of health as the explanatory variable. $$\begin{array}{lrcccc} & \text { Poor } & \text { Fair } & \text { Good } & \text { Excellent } & \text { Total } \\\\\hline \text { Not too happy } & 696 & 1,386 & 1,629 & 732 & 4,443 \\\\\hline \text { Pretty happy } & 950 & 3,817 & 9,642 & 5,195 & 19,604 \\\\\hline \text { Very happy } & 350 & 1,382 & 4,520 & 5,095 & 11,347 \\ \hline \text { Total } & 1.996 & 6.585 & 15.791 & 11.022 & 35.394\end{array}$$

Step by step solution

01

Understand the Problem

We need to determine if there is a relationship between health levels and happiness levels using the provided data table. Here, health is the explanatory variable and happiness is the response variable.

02

Calculate Row Percentages

Convert the counts within each row (happiness level) to percentages. This helps to compare proportions within each happiness level relative to different health levels.

03

Calculate Percentages for 'Not too happy'

For 'Not too happy': \( \text { Poor: } \frac{696}{4443} \times 100 \approx 15.67\% \) \( \text { Fair: } \frac{1386}{4443} \times 100 \approx 31.19\% \) \( \text { Good: } \frac{1629}{4443} \times 100 \approx 36.66\% \) \( \text { Excellent: } \frac{732}{4443} \times 100 \approx 16.48\% \)

04

Calculate Percentages for 'Pretty happy'

For 'Pretty happy': \( \text { Poor: } \frac{950}{19604} \times 100 \approx 4.84\% \) \( \text { Fair: } \frac{3817}{19604} \times 100 \approx 19.47\% \) \( \text { Good: } \frac{9642}{19604} \times 100 \approx 49.18\% \) \( \text { Excellent: } \frac{5195}{19604} \times 100 \approx 26.50\% \)

05

Calculate Percentages for 'Very happy'

For 'Very happy': \( \text { Poor: } \frac{350}{11347} \times 100 \approx 3.08\% \) \( \text { Fair: } \frac{1382}{11347} \times 100 \approx 12.18\% \) \( \text { Good: } \frac{4520}{11347} \times 100 \approx 39.83\% \) \( \text { Excellent: } \frac{5095}{11347} \times 100 \approx 44.91\% \)

06

Interpret the Percentages

Compare the percentages within each happiness level across the different health levels. Observe how the percentage of people being happier changes with better health.

07

Draw Conclusions

Notice the trend: as health improves from 'Poor' to 'Excellent', the percentage of people reported as 'Not too happy' decreases, those reported as 'Pretty happy' have higher percentages for 'Good', and those reported as 'Very happy' have the highest percentages for 'Excellent'. Health likely plays a significant role in happiness.

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Key Concepts

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

Explanatory Variable

In many studies, researchers want to determine the relationship between two variables. An explanatory variable is an independent variable that we think will explain or cause changes in another variable. In this exercise, 'health' is the explanatory variable. We assume that variations in an individual's health (ranging from 'Poor' to 'Excellent') might cause changes in their level of happiness. Therefore, we use health status to explain the levels of happiness. This helps in understanding how improvements or declines in health can impact overall happiness levels.

Response Variable

The response variable is what researchers measure and hope to explain or predict. In the context of this exercise, 'happiness' is the response variable. Happiness is what we observe as a result of changes in the explanatory variable, which is health. We categorize happiness into three levels: 'Not too happy,' 'Pretty happy,' and 'Very happy.' By examining the data in different health categories, we explore how happiness levels respond to changes in health. This can help infer if healthier people tend to be happier or if other patterns exist.

Percentage Calculation

Percentage calculations help in comparing data more meaningfully. Instead of looking at raw numbers, it’s easier to see differences as percentages. To calculate percentages in this context:

• First, take the count for each health level within a happiness category.
• Divide that count by the total number of people in that happiness category.
• Finally, multiply the result by 100 to get a percentage.

This conversion shows the proportion of people in each health level who report each happiness level, making comparisons across categories clearer. For example, if 696 out of 4443 people reporting 'Not too happy' have 'Poor' health, this is approximately 15.67%.

Data Interpretation

After calculating the percentages, interpreting data helps in drawing conclusions about the relationship between health and happiness. Here are key observations:

• For 'Not too happy' individuals, the highest percentage comes from those with 'Good' health. Yet, as health improves to 'Excellent,' this percentage drops.
• 'Pretty happy' individuals show most percentages in 'Good' health but lower in 'Poor' health.
• 'Very happy' people show increasing percentages as health improves, peaking at 'Excellent' health.

This trend suggests a positive relationship between health and happiness. As health status improves, the likelihood of higher happiness also increases. This finding reinforces the idea that better health contributes to greater happiness.