Inference for Regression Slope - Biostatistics

For a regression model predicting blood pressure from age, the output shows $\hat{\beta}_1 = 0.8$ , $SE(\hat{\beta}_1) = 0.25$ , and $t = 3.2$ with $p = 0.003$ . If the researcher constructs a 99% confidence interval for $\beta_1$ , which statement is most likely true?
Answer: The interval will not contain 0 because the test rejects at $\alpha = 0.01$ When you encounter regression output with hypothesis testing results, remember that confidence intervals and hypothesis tests are mathematically linked through the same underlying distribution and significance levels. The key insight is recognizing the relationship between p-values and confidence intervals. Since $p = 0.003 < 0.01$ , we reject the null hypothesis $H_0: \beta_1 = 0$ at the $\alpha = 0.01$ significance level. This means there's sufficient evidence that the true slope is not zero. When a hypothesis test rejects $H_0: \beta_1 = 0$ at significance level $\alpha$ , the corresponding $(1-\alpha) \times 100\%$ confidence interval will not contain zero. Since we reject at $\alpha = 0.01$ , the 99% confidence interval will exclude zero. Choice A incorrectly suggests the interval contains zero because $p > 0.001$ . While true that $p > 0.001$ , the relevant comparison is $p < 0.01$ , which leads to rejection and an interval excluding zero. Choice C misunderstands the relationship between confidence levels and hypothesis testing. A 99% confidence interval corresponds to $\alpha = 0.01$ , and since we reject $H_0$ at this level, the interval won't contain zero. Choice D unnecessarily complicates the situation. The p-value already incorporates the degrees of freedom and critical value information needed for this determination. Study tip: Remember this fundamental relationship: if a two-sided test rejects $H_0: \beta = 0$ at level $\alpha$ , then the $(1-\alpha) \times 100\%$ confidence interval for $\beta$ will not contain zero. This connection appears frequently on biostatistics exams.
A medical researcher tests the effect of a new treatment on recovery time. The regression gives $\hat{\beta}_1 = -3.5$ days per treatment unit with $SE(\hat{\beta}_1) = 1.2$ and $n = 28$ . She wants to show the treatment reduces recovery time. At $\alpha = 0.01$ , what should she conclude?
Answer: Reject $H_0: \beta_1 \geq 0$ because $t = -2.92 < -t_{26,0.01} = -2.479$ When you encounter regression analysis with hypothesis testing, you need to carefully identify what the researcher is trying to prove and set up the appropriate one-tailed or two-tailed test. Since the researcher wants to show the treatment reduces recovery time, she's testing whether the slope is negative. This requires a one-tailed test with $H_0: \beta_1 \geq 0$ versus $H_1: \beta_1 < 0$ . The test statistic is $t = \frac{\hat{\beta}_1}{SE(\hat{\beta}_1)} = \frac{-3.5}{1.2} = -2.92$ with $df = n-2 = 26$ . For a one-tailed test at $\alpha = 0.01$ , the critical value is $-t_{26,0.01} = -2.479$ . Since $t = -2.92 < -2.479$ , we reject $H_0$ and conclude the treatment significantly reduces recovery time. Choice A is correct because it uses the proper one-tailed hypothesis, calculates the test statistic correctly, and applies the right decision rule. Choice B uses the correct hypothesis and test statistic but makes a critical error in the decision rule—it says $-2.92 > -2.479$ , which is mathematically wrong. Choice C incorrectly sets up a two-tailed test ( $H_0: \beta_1 = 0$ ) when the researcher has a specific directional hypothesis. While the conclusion happens to be the same, the approach doesn't match the research question. Choice D also uses the wrong two-tailed setup and incorrectly concludes there's insufficient evidence, missing the significant effect entirely. Study tip: Always match your hypothesis test to the research question. Directional claims like "reduces," "increases," or "improves" require one-tailed tests with appropriately positioned critical regions.
A medical researcher tests $H_0: \beta_1 = 0$ versus $H_1: \beta_1 \neq 0$ for the slope of drug dosage predicting recovery time. With $n = 30$ patients, she obtains $t = -2.85$ . Using $\alpha = 0.01$ , what conclusion should she draw?
Answer: Reject $H_0$ because $|t| = 2.85 > t_{28,0.005} = 2.763$ When testing a regression slope coefficient, you're conducting a two-tailed t-test to determine if there's a significant linear relationship between your variables. Here, the researcher wants to know if drug dosage significantly predicts recovery time. For this hypothesis test, you need the correct degrees of freedom and critical value. With $n = 30$ patients, the degrees of freedom are $df = n - 2 = 28$ (we subtract 2 because we're estimating both the intercept and slope). For a two-tailed test with $\alpha = 0.01$ , you need the critical value at $\alpha/2 = 0.005$ in each tail, giving you $t_{28,0.005} = 2.763$ . Since this is a two-tailed test, you compare the absolute value of your test statistic to the critical value: $|t| = |-2.85| = 2.85 > 2.763$ . Therefore, you reject $H_0$ and conclude there is a significant relationship between dosage and recovery time. Answer A is correct because it uses the proper degrees of freedom (28), the correct critical value for a two-tailed test (0.005), and applies the absolute value correctly. Answer B incorrectly focuses on the sign of the t-statistic, which is irrelevant in a two-tailed test. Answer C uses wrong degrees of freedom (29 instead of 28) and the wrong critical value (0.01 instead of 0.005). Answer D uses the correct degrees of freedom but wrong critical value, leading to an incorrect conclusion. Study tip: For regression slope tests, always remember $df = n - 2$ and use $\alpha/2$ for two-tailed critical values.
A study of advertising spending and sales revenue gives $\hat{\beta}_1 = 4.2$ with $SE(\hat{\beta}_1) = 1.5$ and $n = 18$ . The researcher wants to test if each dollar of advertising increases revenue by more than $3. What is the p-value for this test?
Answer: $P(t_{16} > 0.8) \approx 0.22$ When testing whether a regression coefficient exceeds a specific value, you're conducting a one-tailed hypothesis test. Here, you want to test if advertising increases revenue by more than $3, so your null hypothesis is $H_0: \beta_1 = 3$ and alternative hypothesis is $H_1: \beta_1 > 3$ . The test statistic follows the formula: $t = \frac{\hat{\beta}_1 - \beta_0}{SE(\hat{\beta}_1)} = \frac{4.2 - 3}{1.5} = \frac{1.2}{1.5} = 0.8$ Since you have $n = 18$ observations, your degrees of freedom are $df = n - 2 = 16$ (we subtract 2 because we estimate both the intercept and slope in simple linear regression). Because this is a one-tailed test asking if the coefficient is greater than 3, the p-value is $P(t_{16} > 0.8)$ , which equals approximately 0.22. Answer A is correct because it uses the right degrees of freedom (16), the correct test statistic (0.8), and the proper one-tailed format. Answer B is wrong because it shows a two-tailed test with $|t_{16}|$ , but you're only testing if the coefficient is greater than 3, not different from 3. Answer C uses the wrong degrees of freedom (17 instead of 16) - a common mistake when students forget that simple linear regression costs 2 degrees of freedom. Answer D incorrectly uses 2.8 as the test statistic, possibly from calculating $\frac{4.2}{1.5}$ instead of $\frac{4.2-3}{1.5}$ . Study tip: Always remember that hypothesis tests compare your estimate to the hypothesized value, not to zero, and simple linear regression always uses $n-2$ degrees of freedom.
In a clinical trial, the regression of symptom improvement on drug dose yields $\hat{\beta}_1 = 0.6$ with $SE(\hat{\beta}_1) = 0.3$ and $n = 27$ . A competing drug claims their slope is 0.9. To test if the new drug is inferior (slope less than 0.9) at $\alpha = 0.05$ , what is the test statistic and critical value?
Answer: $t = \frac{0.6-0.9}{0.3} = -1.0$ ; critical value $= -t_{25,0.05} = -1.708$ When testing whether a regression slope is significantly different from a hypothesized value, you're performing a one-sample t-test on the slope coefficient. The key is setting up your null and alternative hypotheses correctly based on what you're trying to prove. Here, you want to test if the new drug is "inferior" (slope less than 0.9), so your alternative hypothesis is $H_a: \beta_1 < 0.9$ and null hypothesis is $H_0: \beta_1 = 0.9$ . This makes it a left-tailed test. The test statistic follows the standard formula: $t = \frac{\hat{\beta}_1 - \beta_{1,hypothesized}}{SE(\hat{\beta}_1)} = \frac{0.6 - 0.9}{0.3} = -1.0$ . The degrees of freedom are $n - 2 = 27 - 2 = 25$ (we lose 2 df in simple linear regression for estimating intercept and slope). For a left-tailed test at $\alpha = 0.05$ , the critical value is $-t_{25,0.05} = -1.708$ . Answer B incorrectly uses the positive critical value $+1.708$ , which would be for a right-tailed test. Answer C completely ignores the hypothesis being tested by calculating $t = \frac{0.6}{0.3}$ , essentially testing if the slope differs from zero rather than from 0.9. Answer D uses the wrong degrees of freedom (26 instead of 25), forgetting that regression analysis requires subtracting 2 from the sample size. Remember: for regression slope tests, always subtract 2 from your sample size for degrees of freedom, and make sure your critical value direction matches your alternative hypothesis (negative for "less than" tests).
In a simple linear regression with 18 degrees of freedom for error, the slope estimate is $\hat{\beta}_1 = 2.4$ with standard error $SE(\hat{\beta}_1) = 0.8$ . What is the 95% confidence interval for the true slope?
Answer: $2.4 \pm 2.101(0.8) = (0.72, 4.08)$ When you're constructing a confidence interval for a regression slope, you need to use the t-distribution because you're estimating the population standard deviation from sample data. The key insight is identifying the correct degrees of freedom to find the appropriate t-critical value. The confidence interval formula is: $\hat{\beta}_1 \pm t_{\alpha/2,df} \times SE(\hat{\beta}_1)$ . Here, you have $\hat{\beta}_1 = 2.4$ and $SE(\hat{\beta}_1) = 0.8$ . For a 95% confidence interval, $\alpha = 0.05$ , so you need $t_{0.025,18}$ . With 18 degrees of freedom, the critical t-value is 2.101. This gives you: $2.4 \pm 2.101(0.8) = 2.4 \pm 1.681 = (0.719, 4.081)$ , which rounds to $(0.72, 4.08)$ . Choice A is correct because it uses the proper t-critical value of 2.101 for 18 degrees of freedom. Choice B incorrectly uses 1.96, which is the z-critical value you'd use if you knew the population standard deviation—a common mistake when students forget that regression problems require the t-distribution. Choice C uses 2.086, which is the t-critical value for 19 degrees of freedom, suggesting confusion about how degrees of freedom are calculated in regression. Choice D uses 2.048, which corresponds to 20 degrees of freedom. Study tip: Always double-check your degrees of freedom in regression problems. For simple linear regression with n observations, error df = n - 2. Then look up the exact t-critical value rather than memorizing approximate values, as small differences significantly affect your interval bounds.
In studying the effect of study hours on exam scores, a researcher obtains the 90% confidence interval for the slope as $(1.2, 4.8)$ . Based on this interval, what can be concluded about testing $H_0: \beta_1 = 0$ versus $H_1: \beta_1 \neq 0$ at $\alpha = 0.10$ ?
Answer: Reject $H_0$ because 0 is not contained in the confidence interval When you encounter confidence intervals for regression slopes, you're dealing with the intimate connection between hypothesis testing and interval estimation. The key insight is that a confidence interval contains all the plausible values for the parameter at your chosen confidence level. The correct reasoning leads to answer A: since the 90% confidence interval (1.2, 4.8) does not contain zero, you can reject $H_0: \beta_1 = 0$ at $\alpha = 0.10$ . This works because there's a direct correspondence between confidence intervals and two-tailed hypothesis tests—if a $(1-\alpha) \times 100\%$ confidence interval doesn't contain the null hypothesis value, you reject that null hypothesis at significance level $\alpha$ . Answer B makes a logical error by suggesting that containing only positive values somehow supports failing to reject the null. The sign of the values isn't the issue—it's whether zero (the null value) is included. Answer C incorrectly assumes you need additional information. The confidence interval already incorporates the sample size and standard error in its construction, giving you everything needed for the hypothesis test conclusion. Answer D reaches the right conclusion but focuses on an incomplete justification. While it's true that the lower bound exceeds zero, the complete reasoning requires recognizing that zero falls outside the entire interval. Study tip: Remember the confidence interval-hypothesis test duality. If your null hypothesis value lies outside a $(1-\alpha) \times 100\%$ confidence interval, reject the null at significance level $\alpha$ . This relationship is your shortcut to hypothesis testing conclusions.
In a regression of weight loss (pounds) on exercise minutes per day, a researcher gets $\hat{\beta}_1 = 0.05$ with $SE(\hat{\beta}_1) = 0.02$ and $n = 30$ . To test if exercise has any effect on weight loss using $\alpha = 0.02$ , what is the rejection region?
Answer: $|t| > t_{28,0.01} = 2.467$ When testing whether a regression coefficient is significantly different from zero, you're conducting a two-tailed hypothesis test. The key elements you need to identify are the degrees of freedom and the appropriate critical value for your significance level. For this regression with $n = 30$ observations, the degrees of freedom are $df = n - 2 = 28$ (we subtract 2 because we estimate both the intercept and slope). Since we're testing "if exercise has any effect" (not specifically positive or negative), this is a two-tailed test with $H_0: \beta_1 = 0$ versus $H_a: \beta_1 \neq 0$ . With $\alpha = 0.02$ for a two-tailed test, you split this between both tails: $\alpha/2 = 0.01$ in each tail. Therefore, you need the critical value $t_{28,0.01} = 2.467$ , and the rejection region is $|t| > 2.467$ . This makes option A correct. Option B uses the wrong degrees of freedom (29 instead of 28) and fails to split alpha for the two-tailed test. Option C has the correct degrees of freedom but incorrectly uses $\alpha = 0.02$ instead of $\alpha/2 = 0.01$ for the critical value lookup. Option D makes the same alpha-splitting error as C and incorrectly specifies a one-tailed test ( $t >$ instead of $|t| >$ ). Remember: for regression coefficient tests, always use $df = n - 2$ , and when testing "any effect" (two-tailed), split your alpha level in half when looking up critical values.
In a pharmaceutical study, 52 patients receive varying doses of a new medication. The regression of symptom improvement score (Y) on dose (X mg) gives $\hat{\beta}_1 = 0.15$ with $SE(\hat{\beta}_1) = 0.06$ . Two researchers disagree about the hypothesis test. Researcher A tests $H_0: \beta_1 = 0$ vs $H_1: \beta_1 > 0$ (one-tailed), while Researcher B tests $H_0: \beta_1 = 0$ vs $H_1: \beta_1 \neq 0$ (two-tailed). If $t_{0.05,50} = 1.676$ and $t_{0.025,50} = 2.009$ , which statement about their conclusions at $\alpha = 0.05$ is correct?
Answer: Both researchers will reject H₀ since the test statistic t = 2.50 exceeds both critical values When you encounter regression hypothesis testing questions, you need to calculate the test statistic and compare it to the appropriate critical value based on whether the test is one-tailed or two-tailed. First, calculate the test statistic: $t = \frac{\hat{\beta}_1 - 0}{SE(\hat{\beta}_1)} = \frac{0.15}{0.06} = 2.50$ Now compare this to each researcher's critical value. Researcher A uses a one-tailed test with $\alpha = 0.05$ , so the critical value is $t_{0.05,50} = 1.676$ . Since $2.50 > 1.676$ , Researcher A rejects $H_0$ . Researcher B uses a two-tailed test with $\alpha = 0.05$ , so the critical value is $t_{0.025,50} = 2.009$ (note the 0.025 because we split alpha between both tails). Since $2.50 > 2.009$ , Researcher B also rejects $H_0$ . Looking at the wrong answers: A incorrectly calculates the test statistic as 1.50 instead of 2.50. B correctly identifies that Researcher A rejects $H_0$ but wrongly concludes that Researcher B doesn't, missing that 2.50 exceeds 2.009. C makes a conceptual error by suggesting the appropriateness of the test determines the statistical conclusion, when both tests yield rejection based purely on the numbers. The correct answer is D because both researchers reject $H_0$ since $t = 2.50$ exceeds both critical values. Study tip: Always calculate the test statistic first, then identify the correct critical value based on the tail direction and significance level. Remember that two-tailed tests split alpha between both tails, requiring a higher critical value.
A biostatistician analyzes the relationship between age (years) and bone density (g/cm²) in 60 postmenopausal women. The regression output shows $\hat{\beta}_1 = -0.012$ with $SE(\hat{\beta}_1) = 0.004$ . When testing $H_0: \beta_1 = 0$ vs $H_1: \beta_1 \neq 0$ at $\alpha = 0.05$ , the critical value is $t_{0.025,58} = 2.002$ . If the researcher instead wants to construct a 90% confidence interval for the slope, what critical value should be used?
Answer: $t_{0.05,58} = 1.672$ For a 90% confidence interval, we need the critical value that leaves 5% in each tail, so we use t₀.₀₅,₅₈ = 1.672. The degrees of freedom remain 58 (n-2 = 60-2). Choice A uses t₀.₁₀,₅₈ which would leave 10% in each tail (80% confidence). Choice C uses the same critical value as the original two-tailed test (95% confidence). Choice D uses t₀.₀₁,₅₈ which corresponds to 98% confidence.
In a regression analysis with $n = 15$ , a researcher obtains $\hat{\beta}_1 = -1.6$ and $SE(\hat{\beta}_1) = 0.4$ . For testing $H_0: \beta_1 \geq 0$ versus $H_1: \beta_1 < 0$ at $\alpha = 0.05$ , what is the critical value and decision?
Answer: Critical value is $-t_{13,0.05} = -1.771$ ; reject $H_0$ because $t = -4.0 < -1.771$ When you encounter hypothesis testing for regression coefficients, you need to carefully identify the type of test (one-tailed vs. two-tailed) and determine the correct degrees of freedom and critical value accordingly. This is a one-tailed test since we're testing $H_0: \beta_1 \geq 0$ versus $H_1: \beta_1 < 0$ . The test statistic is $t = \frac{\hat{\beta}_1 - 0}{SE(\hat{\beta}_1)} = \frac{-1.6}{0.4} = -4.0$ . For simple linear regression with $n = 15$ , the degrees of freedom are $df = n - 2 = 13$ . Since this is a left-tailed test ( $H_1: \beta_1 < 0$ ), the critical value is $-t_{13,0.05} = -1.771$ . We reject $H_0$ because our test statistic $t = -4.0$ falls in the rejection region ( $t < -1.771$ ). Answer A is correct for these reasons. Answer B incorrectly treats this as a two-tailed test, using the positive critical value and absolute value of the test statistic. Answer C uses the wrong degrees of freedom (14 instead of 13) – remember that simple linear regression uses $n-2$ degrees of freedom, not $n-1$ . Answer D uses $\alpha/2 = 0.025$ instead of $\alpha = 0.05$ , which would be appropriate for a two-tailed test but not this one-tailed scenario. Study tip: Always identify whether the alternative hypothesis indicates a one-tailed or two-tailed test, and remember that simple linear regression uses $df = n-2$ because you estimate two parameters (intercept and slope).
A researcher studying the relationship between hours of sleep and reaction time obtains a regression slope of $\hat{\beta}_1 = -15$ milliseconds per hour with $SE(\hat{\beta}_1) = 4$ ms/hour and $n = 20$ . What is the 90% confidence interval for the change in reaction time per additional hour of sleep?
Answer: $-15 \pm 1.734(4) = (-21.94, -8.06)$ milliseconds When constructing confidence intervals for regression coefficients, you need to identify the correct critical value based on your sample size and confidence level. This requires understanding which distribution to use and how degrees of freedom work in regression. The correct approach uses the formula: $\hat{\beta}_1 \pm t_{\alpha/2, df} \times SE(\hat{\beta}_1)$ . With $n = 20$ observations, you have $df = n - 2 = 18$ degrees of freedom (subtracting 2 for the intercept and slope parameters). For a 90% confidence interval, $\alpha = 0.10$ , so $\alpha/2 = 0.05$ . The critical value $t_{0.05, 18} = 1.734$ . Therefore: $-15 \pm 1.734(4) = (-21.94, -8.06)$ . Answer A is correct because it uses the proper t-distribution critical value of 1.734 for 18 degrees of freedom at the 90% confidence level. Answer B incorrectly uses $z_{0.05} = 1.645$ , the standard normal critical value. This is wrong because regression uses the t-distribution, not the normal distribution, especially with small samples. Answer C uses 1.729, which appears to be $t_{0.05, 19}$ (incorrectly using $n-1$ degrees of freedom instead of $n-2$ ). This is a common error when students forget that simple linear regression loses 2 degrees of freedom. Answer D uses 2.101, which is $t_{0.025, 18}$ - the critical value for a 95% confidence interval, not 90%. Study tip: Always remember the "2" rule for simple linear regression: use $n-2$ degrees of freedom and verify your confidence level matches your critical value. Double-check whether the problem calls for t or z distribution.
A regression analysis yields $\hat{\beta}_1 = 1.8$ with $SE(\hat{\beta}_1) = 0.6$ and $n = 22$ . To test $H_0: \beta_1 = 2.5$ versus $H_1: \beta_1 \neq 2.5$ , what is the p-value?
Answer: $P(|t_{20}| > 1.17) \approx 0.26$ When testing hypotheses about regression coefficients, you're conducting a t-test using the sampling distribution of the coefficient estimate. The key framework here is setting up the test statistic and identifying the correct degrees of freedom. To test $H_0: \beta_1 = 2.5$ versus $H_1: \beta_1 \neq 2.5$ , you calculate the t-statistic as: $t = \frac{\hat{\beta}_1 - \beta_{1,0}}{SE(\hat{\beta}_1)} = \frac{1.8 - 2.5}{0.6} = \frac{-0.7}{0.6} = -1.17$ Since this is a two-tailed test ( $H_1: \beta_1 \neq 2.5$ ), the p-value is $P(|t| > 1.17)$ . The degrees of freedom for a simple linear regression are $n - 2 = 22 - 2 = 20$ , giving us $P(|t_{20}| > 1.17) \approx 0.26$ . Answer A is correct because it uses the proper degrees of freedom (20) and correctly formulates the two-tailed test with the absolute value notation. Answer B incorrectly calculates a one-tailed p-value ( $P(t_{20} < -1.17)$ ) instead of the two-tailed test specified by the alternative hypothesis. Answer C uses incorrect degrees of freedom (21 instead of 20). In simple linear regression, you always lose 2 degrees of freedom for the intercept and slope parameters. Answer D incorrectly uses the standard normal distribution (z) instead of the t-distribution. With a sample size of 22, the t-distribution is appropriate, especially since we're working with an estimated standard error. Remember: regression hypothesis tests use t-distributions with $n-2$ degrees of freedom, and two-tailed tests require absolute value notation in the p-value calculation.
A researcher studying the relationship between daily exercise minutes (x) and systolic blood pressure (y) in adults obtains the following regression results: $\hat{y} = 140 - 0.3x$ with $SE(\hat{\beta}_1) = 0.12$ and $n = 25$ . If the researcher wants to test whether exercise significantly reduces blood pressure at $\alpha = 0.05$ , what is the appropriate test statistic?
Answer: $t = \frac{-0.3}{0.12} = -2.50$ When testing whether a regression coefficient is significantly different from zero, you're conducting a one-sample t-test on the slope parameter. The key insight here is understanding what hypothesis you're actually testing and how to construct the appropriate test statistic. The researcher wants to test whether exercise significantly reduces blood pressure, meaning they're testing if the slope $\beta_1$ is significantly less than zero (since negative slopes indicate reduction). The standard t-test formula for regression coefficients is: $t = \frac{\hat{\beta}_1 - \beta_{1,H_0}}{SE(\hat{\beta}_1)}$ Since we're testing against the null hypothesis that $\beta_1 = 0$ , this becomes: $t = \frac{-0.3 - 0}{0.12} = -2.50$ . However, answer A shows the simplified version $t = \frac{-0.3}{0.12} = -2.50$ , which is mathematically equivalent since subtracting zero doesn't change the value. Answer A is correct because it properly calculates the t-statistic using the negative slope coefficient. Answer B incorrectly uses the absolute value of the slope (0.3 instead of -0.3), which would give you the wrong direction for your test. Answer C shows the full formula including "- 0" but this is unnecessarily verbose when the simpler form in A is equivalent. Answer D incorrectly uses a z-statistic instead of a t-statistic, but with small samples (n=25) and unknown population variance, you must use the t-distribution. Study tip: In regression hypothesis testing, always preserve the sign of your coefficient in the test statistic—it tells you the direction of the relationship you're testing.
A researcher wants to determine if there is evidence that each additional year of education increases annual income by more than $2000. The regression gives $\hat{\beta}_1 = 2400$ (in dollars per year) with $SE(\hat{\beta}_1) = 300$ and $n = 35$ . What are the appropriate hypotheses and test statistic?
Answer: $H_0: \beta_1 \leq 2000$ , $H_1: \beta_1 > 2000$ ; $t = \frac{2400-2000}{300} = 1.33$ When testing whether a regression coefficient exceeds a specific threshold, you need to set up hypotheses that directly address the research question. The researcher wants evidence that each additional year of education increases income by more than $2000, which means they're testing whether $\beta_1 > 2000$ . The correct approach uses $H_0: \beta_1 \leq 2000$ and $H_1: \beta_1 > 2000$ . The null hypothesis assumes the increase is $2000 or less, while the alternative tests for increases greater than $2000. The test statistic compares the estimated coefficient to the threshold: $t = \frac{2400-2000}{300} = 1.33$ . This tells us how many standard errors our estimate is above the $2000 threshold. Option B tests whether education has any positive effect on income ( $\beta_1 > 0$ ) rather than whether it exceeds the specific $2000 threshold. While this might seem relevant, it doesn't answer the researcher's actual question about exceeding $2000. Option C uses a two-tailed test ( $H_1: \beta_1 \neq 2000$ ), which would test whether the effect differs from $2000 in either direction. However, the research question specifically asks for evidence of an effect greater than $2000, requiring a one-tailed test. Option D reverses the hypotheses, testing whether the effect is less than $2000 rather than greater than $2000, which contradicts the research objective. Study tip: Always translate the research question into mathematical hypotheses first. Words like "more than," "exceeds," or "greater than" signal one-tailed tests, and your null hypothesis should represent the opposite of what you're trying to prove.
A nutritionist analyzes data from 55 adults to examine how daily fiber intake (g) affects cholesterol reduction (mg/dL) over 6 months. The regression shows $\hat{\beta}_1 = -1.8$ with $SE(\hat{\beta}_1) = 0.6$ . To determine if fiber intake reduces cholesterol by at least 1.5 mg/dL per gram of fiber, the nutritionist performs a one-tailed test. If $t_{0.05,53} = 1.674$ and the calculated test statistic is t = -0.50, what should the nutritionist conclude about the research hypothesis at $\alpha = 0.05$ ?
Answer: Fail to reject H₀ because t = -0.50 > -1.674, indicating insufficient evidence that fiber reduces cholesterol by at least 1.5 mg/dL per gram Testing H₀: β₁ = -1.5 vs H₁: β₁ < -1.5 (one-tailed test for greater reduction). Test statistic: t = (-1.8 - (-1.5))/0.6 = -0.30/0.6 = -0.50. For this one-tailed test, we reject H₀ if t < -1.674. Since -0.50 > -1.674, we fail to reject H₀. Choice A incorrectly applies absolute value. Choice B ignores the role of statistical significance testing. Choice D misunderstands that the test statistic can be either positive or negative depending on the null hypothesis value.
An epidemiologist studies the association between years of smoking and lung function (FEV₁) in 45 current smokers. The regression equation is $\hat{Y} = 3.2 - 0.08X$ where Y is FEV₁ (liters) and X is years of smoking. The standard error of the slope is 0.025. To test whether lung function decreases by more than 0.05 liters per year of smoking, which of the following represents the correct null and alternative hypotheses with the appropriate test statistic?
Answer: $H_0: \beta_1 = -0.05$ , $H_1: \beta_1 < -0.05$ ; $t = \frac{-0.08-(-0.05)}{0.025} = -1.20$ When testing whether a slope coefficient exceeds a specific threshold in regression analysis, you need to carefully translate the research question into proper statistical hypotheses and identify whether you're conducting a one-tailed or two-tailed test. The research question asks whether lung function "decreases by more than 0.05 liters per year." Since we're looking at decrease, the slope should be negative. A decrease of more than 0.05 liters means the slope is more negative than -0.05 (i.e., $\beta_1 < -0.05$ ). This becomes our alternative hypothesis. The null hypothesis states the opposite: that the decrease is 0.05 liters or less, meaning $\beta_1 \geq -0.05$ . However, for computational purposes, we test against the boundary value $\beta_1 = -0.05$ . The t-statistic formula is $t = \frac{\hat{\beta_1} - \beta_{1,null}}{SE(\hat{\beta_1})} = \frac{-0.08 - (-0.05)}{0.025} = -1.20$ . Option A incorrectly uses positive values, ignoring that we're testing decreases (negative slopes). Option B sets up a two-tailed test ( $\neq$ ) when the research question clearly asks about a directional relationship ("more than"). Option C correctly identifies this as a one-tailed test with the right alternative hypothesis, but incorrectly states the null as an inequality rather than the testable point hypothesis. Option D correctly establishes the point null hypothesis $H_0: \beta_1 = -0.05$ with the directional alternative $H_1: \beta_1 < -0.05$ , matching our research question. Study tip: Always translate "more than" or "less than" language into directional (one-tailed) tests, and remember that null hypotheses for t-tests use equality, not inequalities.
A study of 26 patients examines the relationship between medication dose (mg) and symptom improvement score. The 95% confidence interval for the slope is $(-0.5, 2.1)$ . What can be concluded about the effectiveness of the medication?
Answer: There is insufficient evidence that dose affects symptoms at $\alpha = 0.05$ When interpreting confidence intervals for regression slopes, you're essentially testing whether there's a statistically significant relationship between your variables. The key question is: does the interval contain zero? Since the 95% confidence interval for the slope is (-0.5, 2.1), it includes zero. This means we cannot reject the null hypothesis that the true slope equals zero at $\alpha = 0.05$ . In other words, there's insufficient statistical evidence that medication dose affects symptom improvement scores. Answer B correctly identifies this lack of statistical significance. The interval spanning zero indicates we cannot confidently conclude there's any relationship between dose and symptoms. Answer A makes a common error by focusing only on the positive values in the interval. While the interval does contain positive values, it also contains zero and negative values, so we cannot conclude effectiveness based solely on the positive portion. Answer C commits the opposite mistake, focusing only on the negative values. The presence of negative values doesn't prove harm any more than positive values prove benefit when zero is included in the interval. Answer D incorrectly interprets the upper bound as definitive proof of effectiveness. A positive upper bound alone means nothing when the interval includes zero – this suggests the student misunderstands how confidence intervals work. Remember this key principle: when a confidence interval for a slope contains zero, you lack evidence of a significant relationship. Always check whether zero falls within the interval before drawing conclusions about statistical significance.
A researcher studying crop yield versus fertilizer amount obtains the regression output: $\hat{\beta}_1 = 2.3$ kg/hectare per kg fertilizer, $SE(\hat{\beta}_1) = 0.7$ , $n = 21$ . The agricultural guidelines suggest the slope should be at least 2.0. What is the 90% confidence interval and what does it suggest about the guidelines?
Answer: $(1.09, 3.51)$ ; contradicts guidelines since interval contains values below 2.0 When evaluating whether data supports a threshold claim, you need to construct a confidence interval and examine whether it entirely exceeds the threshold value. To find the 90% confidence interval for the slope coefficient, use: $\hat{\beta}_1 \pm t_{\alpha/2} \times SE(\hat{\beta}_1)$ . With $n = 21$ , you have $df = 20$ , so $t_{0.05} = 1.725$ . The interval becomes: $2.3 \pm 1.725(0.7) = 2.3 \pm 1.21 = (1.09, 3.51)$ . The key insight is interpreting what this interval means for the guidelines. Since the guidelines claim the slope should be "at least 2.0," you need the entire confidence interval to be above 2.0 to support this claim. However, this interval extends from 1.09 to 3.51, meaning plausible values include those below 2.0 (like 1.5 or 1.8). This contradicts the guidelines' assertion. Choice A incorrectly states the interval supports the guidelines - it doesn't because part of the interval falls below 2.0. Choice C mischaracterizes this as "insufficient evidence" when the interval actually provides evidence against the guidelines by including values below the threshold. Choice D uses an incorrect critical value (likely $t_{0.025} = 2.086$ ), yielding the wrong interval bounds. Study tip: For threshold claims, the entire confidence interval must be on the correct side of the threshold to provide support. If any part of the interval violates the claim, it contradicts rather than supports the hypothesis.
A researcher fits a linear regression model to examine the relationship between daily exercise minutes (X) and systolic blood pressure (Y) in 50 adults. The fitted model is $\hat{Y} = 145 - 0.8X$ with $SE(\hat{\beta}_1) = 0.25$ . If the researcher wants to test whether each additional minute of exercise is associated with a decrease in systolic blood pressure of more than 0.5 mmHg, what is the appropriate test statistic?
Answer: $t = \frac{-0.8 - (-0.5)}{0.25} = -1.20$ The researcher wants to test if the decrease is more than 0.5 mmHg, so H₀: β₁ = -0.5 vs H₁: β₁ < -0.5 (one-tailed test for more negative slope). The test statistic is t = (β̂₁ - β₁₀)/SE(β̂₁) = (-0.8 - (-0.5))/0.25 = -1.20. Choice B incorrectly uses +0.5 instead of -0.5. Choice C uses the absolute value of the slope estimate. Choice D incorrectly combines absolute value with the wrong null hypothesis value.

Start this quiz