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In a Simple Linear Regression analysis, independent variable is weekly income and dependent variable is weekly consumption expenditure.

Here $95$% confidence interval of regression coefficient, $\beta_1$ is $(.4268,.5914)$.

So i have interpreted as :

"The data provides much evidence to conclude that the true slope of the regression line lies between $.4268$ and $.5914$ at $\alpha=5$% level of significance."

But it is not understandable to those who don't know statistics.

How can i interpret it more generally?

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First in statistical speak:

Our model is $Y_i = \beta_0 + \beta_1 X_i + \varepsilon_i$. This means that $\beta_1$ is the amount that we expect $Y$ to increase by when $X$ increases by 1 [or decrease if $\beta_1 < 0$].

Now in terms of the problem:

In this problem $X$ is weekly income and $Y$ is weekly consumption expenditure so $\hat{\beta_1}$ is our estimate of the amount that weekly consumption expenditure increases for every $\$1$ increase in weekly income. We are 95% confidence that it is between 0.43 and 0.59 [where by "95%" confidence we mean that if we were to collect new data generated from the same distribution then in 19 out of every 20 experiments we'd get $\hat{\beta_1}$ in this interval].

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