Math Core

Every scatterplot of sample data shows some slope, even when the two variables have nothing to do with each other in the population. A significance test for slope asks the key question: is the sample slope far enough from $0$ that chance alone is an unlikely explanation? If so, you have convincing evidence of a real linear relationship between the variables.

Hypotheses

If there is no linear relationship between $x$ and $y$ in the population, then the mean response $\mu_y$ doesn't change as $x$ changes, so the population slope is $\beta = 0$ . That is almost always the null hypothesis.

Hypotheses for a test about slope

$H_0: \beta = 0$

$H_a: \beta > 0 \quad \text{or} \quad H_a: \beta < 0 \quad \text{or} \quad H_a: \beta \ne 0$

Here $\beta$ is the slope of the population regression line relating $x$ to $y$ . Define it in context, and choose the alternative before you look at the data, based on the research question.

A one-sided alternative such as $\beta > 0$ matches a question like "Is there a positive linear relationship?" A two-sided alternative matches "Is there any linear relationship?"

The test statistic

The test follows the same pattern as every $t$ test you've seen: (statistic minus null value) divided by standard error.

t test for the slope

$t = \frac{b - \beta_0}{SE_b}, \qquad df = n - 2$

where $\beta_0$ is the null value (usually $0$ ). The P-value is the probability of getting a $t$ statistic at least this extreme, in the direction of $H_a$ , if $H_0$ is true.

The conditions are the same LINER conditions from the previous lesson: Linear, Independent (10% condition), Normal residuals, Equal SD, and Random.

What the output already tells you

Computer output does most of the arithmetic. In the explanatory variable's row, the T column is $b / SE_b$ and the P column is the P-value for the test of $H_0: \beta = 0$ against the two-sided alternative $H_a: \beta \ne 0$ .

For a two-sided test, use the P-value as printed.
For a one-sided test, if $b$ is on the side that $H_a$ predicts, divide the printed P-value by 2.
If $b$ is on the opposite side from $H_a$ (for example, $b$ is negative but $H_a: \beta > 0$ ), the P-value is greater than $0.5$ and the data give no support to $H_a$ .

Common mistake

The P value in the output is almost always two-sided. Forgetting to halve it for a one-sided test is the most common mistake on these questions. Also, only use the printed T and P when the null value is $0$ . For any other null value, compute $t$ yourself.

A complete significance test

Worked example: Social media and sleep

A school counselor suspects that students who spend more time on social media get less sleep. She selects a random sample of $14$ students at her large high school and records each student's average daily social media use ( $x$ , hours) and average nightly sleep ( $y$ , hours).

x-axis: daily social media use (hours). y-axis: nightly sleep (hours).Open in grapher →

Predictor	Coef	SE Coef	T	P
Constant	7.7608	0.2588	29.99	0.000
Social media	-0.2379	0.0735	-3.24	0.007

S = 0.4563, R-Sq = 46.6%

Do the data give convincing evidence at the $\alpha = 0.05$ level of a negative linear relationship? A residual plot shows no pattern and a dotplot of residuals shows no strong skew.

State. $H_0: \beta = 0$ and $H_a: \beta < 0$ , where $\beta$ is the slope of the population regression line relating nightly sleep to daily social media use for all students at the school. Use $\alpha = 0.05$ .

Plan. $t$ test for the slope.

Linear: The scatterplot is roughly linear and the residual plot shows no curve.
Independent: $14$ is less than $10\%$ of the students at a large high school.
Normal: The dotplot of residuals shows no strong skew or outliers.
Equal SD: The residual plot shows roughly equal spread for all $x$ .
Random: Random sample of students.

Do. From the output, $t = \dfrac{-0.2379 - 0}{0.0735} = -3.24$ with $df = 14 - 2 = 12$ . The printed P-value $0.007$ is two-sided. Since $b$ is negative, as $H_a$ predicts, the one-sided P-value is $0.007 / 2 \approx 0.0035$ .

Conclude. Because the P-value $0.0035$ is less than $\alpha = 0.05$ , reject $H_0$ . There is convincing evidence of a negative linear relationship between daily social media use and nightly sleep for students at this school.

Because this was a random sample, not an experiment, the conclusion is about association. It does not show that social media use causes less sleep.

When the evidence isn't convincing

Worked example: Practice and free throws

A coach takes a random sample of $10$ players in a large youth league and records hours of free-throw practice per week ( $x$ ) and free throws made out of $20$ attempts ( $y$ ). She wants to know whether there is any linear relationship. Assume the conditions are met.

Predictor	Coef	SE Coef	T	P
Constant	11.6485	1.8391	6.33	0.000
Practice	0.4848	0.2588	1.87	0.098

S = 2.3507, R-Sq = 30.5%

$H_0: \beta = 0$ , $H_a: \beta \ne 0$ , where $\beta$ is the slope of the population regression line relating free throws made to weekly practice hours. With $\alpha = 0.05$ :

$t = \dfrac{0.4848}{0.2588} \approx 1.87$ , $df = 8$ , and the two-sided P-value is $0.098$ .

Since $0.098 > 0.05$ , fail to reject $H_0$ . There is not convincing evidence of a linear relationship between practice hours and free throws made for players in this league.

Notice what "fail to reject" does not say. It doesn't prove that $\beta = 0$ ; with only $10$ players, the test may simply lack power to detect a real but modest relationship.

Tip

Switching to a one-sided test after seeing the data would halve the P-value here to about $0.049$ , just under $0.05$ . That is exactly why the alternative must be chosen from the research question before looking at the results.

Testing a nonzero slope

Worked example: Checking a new thermometer

An engineer compares a new digital thermometer ( $y$ ) with a lab standard ( $x$ ) on a random sample of $16$ water baths. If the new thermometer is accurate, the slope should be $1$ . The output gives $b = 1.08$ and $SE_b = 0.035$ . Test $H_0: \beta = 1$ against $H_a: \beta \ne 1$ at $\alpha = 0.05$ . Assume the conditions are met.

The printed T and P test $\beta = 0$ , so compute $t$ yourself:

$t = \frac{1.08 - 1}{0.035} \approx 2.29, \qquad df = 14$

The two-sided P-value is about $0.038$ . Since $0.038 < 0.05$ , reject $H_0$ . There is convincing evidence that the true slope differs from $1$ , so the new thermometer is not tracking the standard perfectly.

Tests and intervals agree

A two-sided test at level $\alpha$ and a $C\%$ confidence interval with $C = 100(1 - \alpha)$ give matching conclusions. If the interval does not contain $0$ , a two-sided test of $H_0: \beta = 0$ rejects $H_0$ . If the interval does contain $0$ , the test fails to reject. The interval gives extra information: a set of plausible values for the size of the slope.

Worked example: Using an interval to decide

A $95\%$ confidence interval for the slope relating a car's engine size (liters) to its highway fuel economy (mpg) is $(-6.8,\ -2.1)$ . What does it say about a test of $H_0: \beta = 0$ versus $H_a: \beta \ne 0$ at $\alpha = 0.05$ ?

The interval does not contain $0$ ; every plausible value of $\beta$ is negative. So the test would reject $H_0$ : there is convincing evidence of a (negative) linear relationship between engine size and highway fuel economy.

Practice

Practice 1

A researcher wants to test whether there is a positive linear relationship between the number of hours a phone has been charging ( $x$ ) and its battery percentage ( $y$ ). Which hypotheses are correct?

Practice 2

Computer output for a regression shows a slope of $0.415$ with a standard error of $0.162$ . Find the test statistic for $H_0: \beta = 0$ . Round to the nearest hundredth.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

A significance test for slope is based on a random sample of $27$ individuals. How many degrees of freedom does the $t$ distribution have?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

Regression output relating $x$ to $y$ shows a slope of $1.36$ in the explanatory variable's row, with P $= 0.084$ . You are testing $H_0: \beta = 0$ against $H_a: \beta > 0$ . What is the P-value for your test?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 5

A test of $H_0: \beta = 0$ versus $H_a: \beta \ne 0$ , relating hours of daylight ( $x$ ) to daily electricity use ( $y$ ) for a random sample of homes, gives a P-value of $0.023$ . Using $\alpha = 0.01$ , which conclusion is correct?

Practice 6

A biologist tests $H_0: \beta = 0$ versus $H_a: \beta > 0$ , where $\beta$ is the slope of the population regression line relating water temperature to the growth rate of a species of algae. Which describes a Type II error in this context?

Practice 7

A theory predicts that the population slope relating $x$ to $y$ is $2$ . From a random sample of $25$ individuals, $b = 2.46$ and $SE_b = 0.19$ . Find the test statistic for $H_0: \beta = 2$ versus $H_a: \beta \ne 2$ . Round to the nearest hundredth.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 8

A $95\%$ confidence interval for the slope relating a runner's weekly mileage ( $x$ ) to her 5K time in minutes ( $y$ ) is $(-0.31,\ 0.07)$ . What can you conclude about a test of $H_0: \beta = 0$ versus $H_a: \beta \ne 0$ at $\alpha = 0.05$ ?