t-interval for slope

t - interval for slope

Population regression line

$$y={\beta }_0+{\beta }_{1}x$$

Sample regression line

$$\hat{y}=b_0+b_{1}x$$

Original SD

$${\sigma }_b=\frac{\sigma }{{\sigma }_x\sqrt{n}}$$

Where

$$S=\sqrt{\frac{\sum (y-\hat{y}{)}^2}{n-2}}$$

and

$$S_x=\sqrt{\frac{\sum (x-\hat{x}{)}^2}{n-1}}$$

General formula

$$b_1\pm t_{n-2}\frac{S}{S_x\sqrt{n-1}}$$

• Where $df=n-2$

Check Condition

Linear

The form of the relationship between x and y is linear. For each value of x, the mean value of y (denoted by $\mu x$) falls on the population regression line ${\mu }_y=\alpha +\beta$

Normal

For each value of x, the distribution of y is approximately normal

Equal SD

For each value of x, the standard deviation of y (denoted by $\sigma$) is the same

Random

The data come from a random sample from the population of interest or a randomized experiment

10%

When sampling without replacement, $n\le \frac{1}{10}N$

t - test for slope

State

We use a t-interval for $\beta$ (slope of population regression line) to construct a $90\%$
Let $x$ be the miles driven, let $y$ be the price.
90% of the slope of the popuation regression line relating y = price to x = miles driven for used

Check conditions

Linear

According to the graphs, the form of the relationship between $x$ and $y$ is linear. For each value of $x$, the mean value of $y$ (denoted by $\mu x$) falls on the population regression line ${\mu }_y=\alpha +\beta$

Normal

For each value of $x$, the distribution of $y$ is approximately normal

Equal SD

For each value of $x$, the standard deviation of $y$ (denoted by $\sigma$) is the same

Random

The data come from a random sample from the population of interest or a randomized experiment

10%

When sampling without replacement, $n\le \frac{1}{10}N$

Do

$$CI=b_1\pm t_{n-2}\frac{S}{S_x\sqrt{n-1}}$$

Conclusion

$$t_{n-2}=\frac{b-{\beta }_0}{S{E}_b}$$