**Study on Education and Income**A survey was conducted on 700 Californians older than 30 years of age. The study aims to understand the relationship between years of education and yearly income in dollars. In this context, the response variable is income in dollars, and the explanatory variable is years of education.A simple linear regression model was employed, and the output from the statistical software R is provided below:```lm(formula = Income ~ Education, data = CA)```### Model Output#### Coefficients:| | Estimate | Std. Error | t value | Pr(>|t|) ||-----------------|----------|------------|---------|-------------|| (Intercept) | 25200.25 | 1488.94 | 16.93 | 3.08e-10 ***|| Education | 2905.35 | 112.61 | 25.80 | 1.49e-12 ***|#### Other Statistics:- **Residual Standard Error**: 32400 on 698 degrees of freedom- **Multiple R-squared**: 0.7602**Explanation of the Output:**- **Intercept (25200.25)**: This value suggests that an individual with zero years of education would have an expected yearly income of $25,200.25.- **Education (2905.35)**: This coefficient indicates that for each additional year of education, an individual's income is expected to increase by $2,905.35, on average.- **Standard Error**: Represents the standard deviation of the error term for the coefficient estimates.- **t value**: The test statistic for the null hypothesis that the coefficient is equal to zero.- **Pr(>|t|)**: The p-value associated with the t-statistic; a very small p-value (typically < 0.05) indicates that the coefficient is statistically significant.- **Residual Standard Error (32400)**: Provides an estimate of the standard deviation of the error term.- **Multiple R-squared (0.7602)**: Represents the proportion of the variance in the dependent variable (Income) that is predictable from the independent variable (Education). A value of 0.7602 suggests a strong relationship.Overall, the regression analysis indicates a significant positive relationship between years of education and yearly income, with higher levels of education correlating with higher income ### Regression Analysis and Hypothesis TestingThe population regression model is given by:\[ \text{Income}_i = \beta_0 + \beta_1 \text{Education}_i + \epsilon_i \]We are interested in testing whether Education is a significant predictor or explanatory variable in explaining Income levels.**Objective:**Determine the null and alternative hypothesis for testing the significance of the predictor variable, Education.**Hypotheses:**- **Null Hypothesis ($ H_0 $)**: $\beta_1 = 0$- **Alternative Hypothesis ($ H_a $)**: $\beta_1 \neq 0$**Question:**Which of the following represents our null and alternative hypothesis for this test?1. $ H_0 : \epsilon = 0 $ vs $ \epsilon \neq 0 $2. $ H_0 : \beta_1 = 0 $ vs $ \beta_1 \neq 0 $3. $ H_0 : \beta_0 = 0 $ vs $ \beta_0 \neq 0 $4. $ H_0 : \text{income} = 0 $ vs $ H_a : \text{income} \neq 0 $### Explanation:To test if Education ($\text{Education}_i$) is a significant predictor of Income ($\text{Income}_i$), we focus on the slope coefficient, $\beta_1$. The appropriate hypothesis test involves testing whether $\beta_1 = 0$ (no effect of Education on Income) against the alternative that $\beta_1 \neq 0$ (a significant effect of Education on Income). Thus, the correct choice is:\[ H_0 : \beta_1 = 0 \ \text{vs} \ \beta_1 \neq 0 \]This test examines if the coefficient $\beta_1$ is statistically different from zero, indicating the importance of Education as an explanatory variable for Income.

The simple linear regression model is:Here, the dependent variable is income and the independent…

Answered: The population regression model is…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

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I’m interested in seeing whether or not there is a relationship between the number of hours someone sleeps and traffic accidents. I gather data from the Department of Transportation and get the following data: X (hours of sleep) Y (Traffic accidents) 5 2 5 1 3 4 4 4 2 6 6 3 8 1 9 1 5 0 6 4 Using the above data, create a linear regression equation to predict traffic accidents from hours of sleep. What is the Pearson correlation coefficient? Is it strong or weak? Positive or negative? What is the linear regression equation? (in the form of Y’ = bX + a) How many traffic accidents might we predict with someone that gets 0 hours of sleep? How many traffic accidents might we predict with someone that gets 11 hours of sleep?
Data from 10 Western European countries: summary statistics show that the mean percent of teens that use marijuana is 23.9 with a standard deviation of 15.6. The mean percent of teens that use other drugs is 11.6 with a standard deviation of 10.2. The correlation coefficient is r= 0.93 Find the equation of the least squares regression line considering Marijuana Use as the Explanatory variable and Other Drug Use as the Response variable. O a. ý = 7.43 + 1.42x O b. ý = - 1.42 -0.61x %3D Oc = -2.98 +0.61x %3D O d. ý = 6.10 +2.98x
28
The following table gives information on the amount of sugar (in grams) and the calorie count in one serving of a sample of 13 varieties of Kellogg's cereal. Sugar (grams) 4. 15 12 11 8. 8 14 20 13 Calories 140 200 150 110 140 80 200 100 120 190 || 190 110 120 %3D The predictive regression equation of the number of calories on the amount of sugar is y = 97.555 + 4.730x, where x is amount of sugar (in grams) and y is calories. Estimate the calorie count for a cereal with 54 grams of sugar per serving. Round your answer to the nearest integer. calories
When testing for bl in a regression model, the null hypothesis (2-tailed) is b1=0. Why? O Then the alternative hypothesis can be b0=1 This indicates that bl=b2 This indicates that bl=b0 A change in x does not result in a change in y
A researcher is interested in examining the relationship between spousal abuse and child abuse. Specifically, they are interested in determining whether there is a predictive relationship between spousal abuse and child abuse in 5 county social services offices. Calculate the linear regression line for the following data. Note you have already calculated the first step to this analysis (Pearson's Correlation)

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