Practice Inference for Regression in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

Using hypothesis tests and confidence intervals to draw conclusions about the true population slope β1\beta_1 of the linear relationship y=β0+β1x+εy = \beta_0 + \beta_1 x + \varepsilon, based on sample data.

You computed a sample regression line with slope b=2.3b = 2.3. But is the true population slope actually different from zero? Maybe there's really no linear relationship and you just got a slope by chance. The regression t-test asks: 'Is my sample slope far enough from zero that it's unlikely to have occurred by random variation alone?'

Showing a random 20 of 50 problems.

Example 1

medium
Why is the slope SE inversely related to the spread of the xx values?

Example 2

easy
In regression inference with sample size n=20n = 20, how many degrees of freedom does the t-test for the slope use?

Example 3

challenge
A regression on a population shows true slope β=2.0\beta = 2.0. A sample of n=20n = 20 gives b=0.5b = 0.5, SEb=1.0SE_b = 1.0. Will a 95% CI capture β=2.0\beta = 2.0? (Use t∗=2.093t^* = 2.093.)

Example 4

challenge
A slope test has df=n−2=8df = n - 2 = 8. A student mistakenly used df=n−1=9df = n - 1 = 9, getting a slightly different p-value. Explain the correct df and why regression loses one more degree of freedom than a one-sample mean.

Example 5

medium
A regression of test scores on study hours gives b=5.0b = 5.0, SEb=1.0SE_b = 1.0, n=25n = 25. Construct a 95% CI (t0.025,23∗=2.069t^*_{0.025, 23} = 2.069) and interpret in context.

Example 6

medium
A regression output shows: slope b=2.5b=2.5, SEb=0.8SE_b=0.8, n=30n=30. Test H0:β=0H_0: \beta=0 vs Ha:β≠0H_a: \beta \neq 0 at α=0.05\alpha=0.05 using a t-test.

Example 7

medium
For b=1.5b = 1.5, SEb=0.5SE_b = 0.5, n=20n = 20, find the 95% CI (t0.025,18∗=2.101t^*_{0.025, 18} = 2.101).

Example 8

easy
A confidence interval for the slope is (1.2,3.8)(1.2, 3.8). Does it provide evidence of a nonzero slope at the matching significance level?

Example 9

medium
A residual plot for a regression shows a clear curve. Why does this matter before doing slope inference?

Example 10

hard
A linear regression is fit on bivariate data; software output: slope b=0.15b = 0.15, p-value =0.20= 0.20, R2=0.10R^2 = 0.10, n=100n = 100. State whether the slope is statistically significant at α=0.05\alpha = 0.05 and interpret.

Example 11

medium
A scatterplot shows a clear curved pattern, but a linear regression gives slope p-value <0.001< 0.001. What is the danger of trusting this inference?

Example 12

medium
A slope estimate is b=0.0b = 0.0 exactly with SEb=0.5SE_b = 0.5. What is the slope t-statistic and likely conclusion?

Example 13

easy
What is the null hypothesis when testing whether a linear relationship exists between xx and yy?

Example 14

medium
If t=−3.0t = -3.0 on df=18df = 18, find the two-sided p-value (approximate).

Example 15

easy
If you double the sample size, what happens (approximately) to SEbSE_b?

Example 16

hard
Explain why a significant slope does NOT imply a causal relationship between xx and yy.

Example 17

easy
A slope CI of (−0.5,2.0)(-0.5, 2.0) is reported. What does it suggest about the slope being zero?

Example 18

medium
A slope test yields t=1.2t = 1.2 on df=15df = 15, p-value =0.25= 0.25, at α=0.05\alpha = 0.05. Conclude.

Example 19

hard
Construct a 95% confidence interval for the slope β\beta given: b=1.8b=1.8, SEb=0.5SE_b=0.5, n=25n=25, and t0.025,23∗=2.069t^*_{0.025,23}=2.069.

Example 20

medium
A two-sided slope test has t=2.5t = 2.5 on df=8df = 8. For a one-sided test (slope >0> 0) with the same data, how does the p-value change?
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