9.4 Carrying Out a Test for the Slope of a Regression Model

• The alternative hypothesis for the slope states that it is not zero
• The significance level represents the maximum probability of rejecting the null hypothesis when it is true.


  True
• A significance level of $\alpha =$$0.05$ implies a 5% chance of making a Type I error
• What is the alternative hypothesis for the slope in linear regression?

  $H_{a}: \beta \neq 0$
• A significance level of $\alpha =$$0.01$ requires stronger evidence to reject the null hypothesis than $\alpha =$$0.05$.


  True
• A significance level of 0.05 corresponds to a 5% probability of a Type I error.
• The significance level ($\alpha$) represents the maximum probability of rejecting $H_{0}$ when it is actually true.
• Steps in testing the slope of a regression model

  1️⃣ State the hypotheses

  2️⃣ Set the significance level

  3️⃣ Calculate the test statistic

  4️⃣ Determine the p-value

  5️⃣ Make a decision about the null hypothesis

  6️⃣ State the conclusion
• The null hypothesis in linear regression for the slope is $H_{0}: \beta =$$0$


  True
• A commonly used significance level is 0.05
• What happens if the p-value is less than the significance level?

  Reject the null hypothesis
• What type of evidence is required when using a significance level of $\alpha =$$0.01$?


  Strong evidence
• What is the significance level used to decide in hypothesis testing?

  The maximum Type I error
• What is the interpretation of a significance level of \alpha = 0.10</latex>?

  Weaker evidence needed
• A significance level of 0.01 requires stronger evidence to reject the null hypothesis compared to a significance level of 0.05.


  True
• Match the significance level with its interpretation:

  $0.05$ ↔️ Moderate evidence needed

  $0.01$ ↔️ Strong evidence needed

  $0.10$ ↔️ Weaker evidence needed
• In an example regression, what is the test statistic $t$ if $\hat{\beta} =$$2.5$ and SE(\hat{\beta}) = 0.8</latex>?


  3.125
• If $t =$$2.75$, $n =$$30$, $\alpha =$$0.05$, and the p-value is $< 0.05$, we reject the null hypothesis.


  True
• If $\alpha =$$0.05$ and the p-value is 0.07, we fail to reject the null hypothesis.


  True
• Common significance levels include $0.05$, $0.01$, and $0.10$ with corresponding probabilities of Type I error of 5%, 1%, and 10%.
• If \hat{\beta} = 2.5</latex> and $SE(\hat{\beta}) =$$0.8$, the test statistic t is 3.125.
• Rejecting the null hypothesis means there is sufficient evidence to support an alternative hypothesis.

  True
• What does the null hypothesis state in linear regression for the slope?

  The slope is zero
• What is the alternative hypothesis for the slope in linear regression?

  $H_{a}: \beta \neq 0$
• Match the significance level with its interpretation:

  $0.05$ ↔️ Moderate evidence needed

  $0.01$ ↔️ Strong evidence needed

  $0.10$ ↔️ Weaker evidence needed
• The significance level is the probability of making a Type I error.


  True
• The null hypothesis for the slope in linear regression is $H_{0}: \beta =$$0$.


  True
• Match the significance level with its interpretation:

  $0.05$ ↔️ Moderate evidence needed

  $0.01$ ↔️ Strong evidence needed

  $0.10$ ↔️ Weaker evidence needed
• What does a significance level of 0.05 indicate in hypothesis testing?

  Moderate evidence needed
• What is the significance level used to decide in hypothesis testing?

  Rejection threshold
• If a p-value is 0.07 and \alpha = 0.05</latex>, we fail to reject the null hypothesis.


  True
• Steps to find the p-value using a t-table:

  1️⃣ Determine the degrees of freedom

  2️⃣ Locate the test statistic in the t-table

  3️⃣ If two-tailed, multiply the p-value by 2

  4️⃣ Compare the p-value to $\alpha$
• What is the null hypothesis in testing the slope of a linear regression?

  $H_{0}: \beta =$$0$
• The significance level represents the maximum probability of rejecting the null hypothesis when it is actually true.

  True
• A significance level of $0.05$ requires moderate evidence to reject the null hypothesis.
• A significance level of $0.01$ requires stronger evidence than 0.05</latex> to reject the null hypothesis.


  True
• The p-value is the probability of observing a test statistic as extreme or more extreme than the calculated one, assuming the null hypothesis is true.


  True
• When stating the conclusion of a hypothesis test, it is sufficient to indicate whether you reject or fail to reject the null hypothesis without further explanation.

  False
• The formula for calculating the test statistic $t$ in linear regression is $t =$$\frac{\hat{\beta} - 0}{SE(\hat{\beta})}$, where $\hat{\beta}$ is the estimated slope.
• The degrees of freedom in a t-test are calculated as $df =$$n - 2$, where $n$ is the sample size.