Use a simple regression model to test the hypothesis H0: β1=0 versus H1: β1 ≠0 with α=0.05, give

step 1 So we want to find whether we are going to be rejecting or not for the slope being zero or the s

Use a simple regression model to test the hypothesis H0:...

Use a simple regression model to test the hypothesis $$ H_0: \beta_1=0 $$ versus $$ H_1: \beta_1 \neq 0 $$ with $\alpha=0.05$, given the following regression statistics. a. The sample size is $35, S S T=100,000$, and the correlation between $X$ and $Y$ is 0.46 . b. The sample size is $61, S S T=123,000$, and the correlation between $X$ and $Y$ is 0.65 . c. The sample size is $25, S S T=128,000$, and the correlation between $X$ and $Y$ is 0.69 .

Use a simple regression model to test the hypothesis
$$
H_0: \beta_1=0
$$
versus
$$
H_1: \beta_1 \neq 0
$$
with $\alpha=0.05$, given the following regression statistics.
a. The sample size is $35, S S T=100,000$, and the correlation between $X$ and $Y$ is 0.46 .
b. The sample size is $61, S S T=123,000$, and the correlation between $X$ and $Y$ is 0.65 .
c. The sample size is $25, S S T=128,000$, and the correlation between $X$ and $Y$ is 0.69 .

Key Concepts

Simple Linear Regression

Simple linear regression is a statistical method that models the relationship between a dependent variable and an independent variable by fitting a linear equation to the observed data. It is used to estimate the slope and intercept, where the slope indicates how much the dependent variable changes on average with a one-unit change in the independent variable.

Hypothesis Testing for Regression Coefficients

Hypothesis testing for regression coefficients involves testing a specific claim about the population parameter, typically the slope. The null hypothesis usually states that there is no linear relationship (i.e., the slope is zero), while the alternative hypothesis suggests that there is a relationship (i.e., the slope is not zero). This testing is fundamental for assessing the statistical significance of the predictors in the model.

t-Test Statistic in Regression Analysis

The t-test statistic is used to determine whether a regression coefficient is significantly different from zero. It is calculated as the estimated coefficient divided by its standard error. Under the null hypothesis, this statistic follows a t-distribution with degrees of freedom equal to the number of observations minus the number of parameters estimated, allowing one to compute p-values and assess significance.

Correlation Coefficient

The correlation coefficient measures the strength and direction of the linear relationship between two variables. In the context of regression analysis, it can provide insight into the degree of association between the independent and dependent variables, and it is related to the regression slope when both variables are standardized.

Degrees of Freedom

Degrees of freedom in regression analysis refer to the number of independent pieces of information available for estimating parameters. In simple linear regression, it is typically calculated as the sample size minus two, accounting for the estimation of both the slope and the intercept. This concept is essential for accurately determining the critical values of the t-distribution when testing hypotheses.

Significance Level

The significance level, usually denoted by alpha, is the threshold for rejecting the null hypothesis. It represents the probability of making a Type I error, which is the incorrect rejection of a true null hypothesis. In hypothesis testing for regression coefficients, the chosen significance level (such as 0.05) determines the critical region for the test statistic.

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