A random sample of 79 patients from a walk-in clinic records the low- and high-density cholesterol levels of the patients, labeled LDL and HDL, respectively. LDL is often referred to as "bad" cholesterol, while HDL is often referred to as "good" cholesterol. A researcher is interested in fitting the following linear regression curve to LDL and HDL levels:

LDL_i = α+ β× HDL_i + Ɛ_i
Where the deviations Ɛ_i are assumed to be independent and Normally distributed with mean 0 and standard deviation σ. This model was fit to the data using the method of least squares. The following results were obtained from statistical software:
$\begin{array} { | c | c | c | } \hline \text { Variable } & \text { Estimate } & \begin{array} { c } \text { Standard Error of } \\\text { Estimate }\end{array} \\\hline \text { Constant } & - 25 & 3.000 \\\text { HDL } & 0.75 & 0.025 \\\hline\end{array}$ r² = 0.810, s = 50.
Suppose the researchers test the following hypotheses:
H₀: β₁ = 0, H_a: β₁ > 0
What is the P-value of the test?

Question

A random sample of 79 patients from a walk-in clinic records the low- and high-density cholesterol levels of the patients, labeled LDL and HDL, respectively. LDL is often referred to as "bad" cholesterol, while HDL is often referred to as "good" cholesterol. A researcher is interested in fitting the following linear regression curve to LDL and HDL levels:

LDL_i = α+ β× HDL_i + Ɛ_i
Where the deviations Ɛ_i are assumed to be independent and Normally distributed with mean 0 and standard deviation σ. This model was fit to the data using the method of least squares. The following results were obtained from statistical software:

\begin{array} { | c | c | c | } \hline \text { Variable } & \text { Estimate } & \begin{array} { c } \text { Standard Error of } \\\text { Estimate }\end{array} \\\hline \text { Constant } & - 25 & 3.000 \\\text { HDL } & 0.75 & 0.025 \\\hline\end{array}

r² = 0.810, s = 50.
Suppose the researchers test the following hypotheses:
H₀: β₁ = 0, H_a: β₁ > 0
What is the P-value of the test?

Quizplus · Accepted Answer

The null hypothesis is that there is no relationship between LDL and HDL levels (β=0), while the alternative is that there is a positive relationship (β>0). The test statistic for this hypothesis is the t-statistic, which is calculated as: t = (β_hat - 0) / (s / √n), where β_hat is the estimated value of β from the regression, s is the standard error of the regression, and n is the sample size.

From the output, we see that β_hat = 0.392 and s = 50. Therefore, t = (0.392 - 0) / (50 / √79) = 2.204. The degrees of freedom for this test are df = n-2 = 77.

Using a t-distribution table (or software), we find that the P-value for this test is less than 0.01, which means that we reject the null hypothesis at the 1% level of significance. Therefore, we can conclude that there is strong evidence of a positive relationship between LDL and HDL levels.

A Random Sample of 79 Patients from a Walk-In Clinic