### Sleep and Happiness Correlation Study**Objective:**To determine if there is a correlation between the amount of sleep a person gets and their level of happiness.**Method:**A sample of 18 individuals was randomly selected. Their average sleep duration over the course of three weeks was recorded. Subsequently, they were asked to rate their happiness on a scale from 1 to 10, where 1 indicates being miserable, 5 is mediocre, and 10 is joyful.**Results:**#### Scatter Plot Explanation:The scatter plot visualizes the relationship between the hours of sleep and the happiness score.- **X-axis:** Represents the number of hours of sleep.- **Y-axis:** Represents the happiness score.- Each dot on the scatter plot corresponds to the data point from one individual in the sample.From the plot, it can be observed that there is a positive correlation: individuals reporting more hours of sleep tend to have higher happiness scores.#### Statistical Analysis:- **Regression Equation:** \[ \text{Happiness Score} = -3.8601533 + 1.4482759 \times \text{Hours of Sleep} \] This equation suggests that, on average, each additional hour of sleep is associated with an increase of approximately 1.448 in the happiness score.- **Sample Size:** The sample size for this study is 18.- **Correlation Coefficient (R):** \[ R = 0.93478344 \] This high correlation coefficient indicates a strong positive relationship between the hours of sleep and the happiness score.- **R-Squared (R²):** \[ R^2 = 0.87382007 \] This value implies that approximately 87.38% of the variance in happiness scores can be explained by the number of hours of sleep.The findings from this study support the theory that increased sleep is associated with higher levels of happiness.### Conclusion:Getting more sleep may contribute significantly to a person's overall happiness. This data-backed insight can be valuable for both individuals seeking to improve their well-being and professionals in the mental health field.### Discussion Points:- What are potential limitations of this study?- How can this data be used to inform public health recommendations?- What further research could be conducted to build on these findings? ### Understanding the Relationship Between Sleep and HappinessBelow are a series of questions designed to help students analyze the relationship between the number of hours of sleep and happiness scores. The analysis is based on a scatterplot and regression analysis data which are necessary for solving the questions. **Questions and Explanations:**a. **Which is the explanatory and which is the response variable?** - The explanatory variable is the number of hours of sleep, and the response variable is the happiness score.b. **According to the scatterplot above, what were the happiness scores for the three people who slept 5.5 hours?** - The specific happiness scores for individuals who slept 5.5 hours can be determined by examining the scatterplot. Unfortunately, without the visual, the exact scores can't be read here directly.c. **According to the scatterplot, does there appear to be a linear relationship between the variables? Explain how you know.** - To ascertain if a linear relationship exists, you would look for a pattern in the scatterplot where data points form a line or a near-line trend. If the plot depicts a clear upward or downward trend, it likely indicates linearity.d. **What is the strength and direction of the correlation? Explain how you know.** - The strength of the correlation can be identified by how closely the data points cluster around the line of best fit on the scatterplot. A strong correlation means points are close to the line, whereas a weak correlation means they are more spread out. The direction (positive or negative) is determined by the slope of the line; upward indicates a positive correlation and downward indicates a negative one.e. **How much variability in the model for happiness is due to the number of hours of sleep?** - This typically refers to the R-squared value in the regression analysis, which explains the proportion of the variance in the response variable (happiness) that is predictable from the explanatory variable (hours of sleep).f. **According to the results of the test, would you say that the amount that a person sleeps is related to their happiness? Explain, using the data and results from the test. Answers without explanation using the statistics collected will not be given credit.** - By analyzing the p-value and correlation coefficient in context, one could determine if there is a statistically significant relationship between sleep and happiness. A low p-value (typically less than 0.05) along with a

"Since you have posted a question with multiple sub-parts, we will solve first three sub-parts for…

a. Which is the explanatory and which is the response variable? b. According to the scatterplot above, what were the happiness scores for the three people who slept 5.5 hours? c. According to the scatterplot, does there appear to be a linear relationship between the variables? Explain how you know. d. What is the strength and direction of the correlation? Explain how you know. How much variability in the model for happiness is due to the number of hours of sleep? f. According to the results of the test, would you say that the amount that a person sleeps is related to their happiness? Explain, using the data and results from the test. Answers without explanation using the statistics collected will not be given credit. g. Use the regression formula above to predict the following. Show your work to three decimal places: i. Number of hours of sleep=8.0, Happiness score=? ii. Number of hours of sleep=5.5. Happiness score=?

a. Which is the explanatory and which is the response variable? b. According to the scatterplot above, what were the happiness scores for the three people who slept 5.5 hours? c. According to the scatterplot, does there appear to be a linear relationship between the variables? Explain how you know. d. What is the strength and direction of the correlation? Explain how you know. How much variability in the model for happiness is due to the number of hours of sleep? f. According to the results of the test, would you say that the amount that a person sleeps is related to their happiness? Explain, using the data and results from the test. Answers without explanation using the statistics collected will not be given credit. g. Use the regression formula above to predict the following. Show your work to three decimal places: i. Number of hours of sleep=8.0, Happiness score=? ii. Number of hours of sleep=5.5. Happiness score=?

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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