Question

to test the effectiveness of a new treatment, a clinician measures cholesterol levels (in mg/dl) of 9 patients before and after treatment. the mean difference is 12, and the standard deviation of the differences is 6. what is the p - value for this test?
.0001
.0003
.0005
.05

Explanation:

Step1: Calculate the t - statistic

The formula for the t - statistic in a paired - t test is \(t=\frac{\bar{d}}{s_d/\sqrt{n}}\), where \(\bar{d}\) is the mean difference, \(s_d\) is the standard deviation of the differences, and \(n\) is the number of pairs.
Given \(\bar{d} = 12\), \(s_d=6\), \(n = 9\)

$$t=\frac{12}{6/\sqrt{9}}=\frac{12}{6/3}=\frac{12}{2}=6$$

Step2: Determine the degrees of freedom

The degrees of freedom for a paired - t test is \(df=n - 1\). Since \(n = 9\), \(df=9-1 = 8\)

Step3: Find the p - value

Using a t - distribution table or a statistical software (for a one - tailed test, assuming the treatment is expected to decrease cholesterol levels), for \(t = 6\) and \(df=8\)
Looking up in the t - table or using a calculator (e.g., in R: 1 - pt(6,8)), the p - value is approximately \(0.0003\)

Answer:

\(0.0003\)