91Ó°ÊÓ

A doctor diagnoses a patient as hypertensive and prescribes an antihypertensive medication. To assess the clinical status of the patient, the doctor takes \(n\) replicate blood-pressure measurements before the patient starts the drug (baseline) and \(n\) replicate blood-pressure measurements 4 weeks after starting the drug (follow-up). She uses the average of the \(n\) replicates at baseline minus the average of the \(n\) replicates at follow-up to assess the clinical status of the patient. She knows, from previous clinical experience with the drug, that the mean diastolic blood pressure (DBP) change over a 4-week period over a large number of patients after starting the drug is \(5.0 \mathrm{mm}\) Hg with variance \(33 / n,\) where \(n\) is the number of replicate measures obtained at both baseline and follow-up. The physician also knows that if a patient is untreated (or does not take the prescribed medication), then the mean DBP over 4 weeks will decline by 2 mm Hg with variance 33/n. What is the probability that an untreated subject will decline by at least \(5 \mathrm{mm}\) Hg if 1 replicate measure is obtained at both baseline and follow-up?

Step by step solution

01

Define the Context

We need to find the probability that an untreated subject will experience a decline in mean diastolic blood pressure (DBP) of at least 5 mm Hg over a 4-week period, given that we have one replicate measurement (\(n = 1\)).

02

Determine the Mean and Variance

From the problem, if untreated, the mean change in DBP is a decline by 2 mm Hg. The variance for one replicate is given as \( \frac{33}{n} \). Since \(n = 1\), the variance is 33.

03

Calculate Standard Deviation

The standard deviation is the square root of the variance. Thus, \( \sigma = \sqrt{33} \approx 5.74456 \).

04

Calculate the Z-Score for a Decline of 5 mm Hg

The Z-score formula is \( Z = \frac{X - \mu}{\sigma} \), where \(X\) is the observed value, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. Substituting the values: \[ Z = \frac{5 - 2}{\sqrt{33}} = \frac{3}{5.74456} \approx 0.522 \]

05

Find the Probability Using the Z-Score

Using a Z-table, find the probability corresponding to a Z-score of 0.522. This is approximately 0.6997, meaning there's a 69.97% probability a value will be less than 5 mm Hg. Thus, the probability of a decline of at least 5 mm Hg is \(1 - 0.6997 = 0.3003\), or 30.03%.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Blood Pressure Measurement

Blood pressure measurement is a vital practice in healthcare. It helps monitor and assess the cardiovascular health of individuals. Blood pressure has two components: systolic and diastolic pressure. Systolic is when the heart beats, while diastolic is when the heart rests between beats.
Regular monitoring of blood pressure can aid in identifying conditions like hypertension. Accurate readings require a calm environment, a properly sized cuff, and the patient being at rest. Consistent measurement practices ensure reliability and allow for effective tracking of any changes over time.

• Use a properly calibrated sphygmomanometer.
• Ensure patients are seated comfortably, typically with arm level with the heart.
• Take multiple readings for accuracy.

Clinical Assessment

Clinical assessment is a comprehensive evaluation of a patient's health. This process includes gathering medical history, conducting a physical exam, and using diagnostic tests. In our context, blood pressure measurement is one of the key components of clinical assessment.
By assessing blood pressure before and during treatment, healthcare professionals can evaluate the effectiveness of interventions like antihypertensive drugs. The clinician monitors changes over a designated period to ensure the patient's condition is improving as expected.

• Understand patient history.
• Evaluate baseline and follow-up measurements.
• Use data to guide treatment decisions.

Z-Score Calculation

Z-score calculation is a statistical method used to understand the relative position of a data point within a distribution. In the context of our exercise, Z-scores help assess whether changes in blood pressure are statistically significant, considering normal variations.
The formula for calculating a Z-score is:\[ Z = \frac{X - \mu}{\sigma} \]where \(X\) is the observed data, \(\mu\) is the mean of the distribution, and \(\sigma\) is the standard deviation.By calculating the Z-score, clinicians can interpret how the observed blood pressure change compares to the expected change. This is crucial for identifying significant outcomes and making informed clinical decisions.

Variance and Standard Deviation

Variance and standard deviation are statistical measures that describe the spread or dispersion of a set of data. Variance quantifies the extent to which each number in the set differs from the mean. It's calculated as the average of the squared differences from the mean.
Standard deviation, on the other hand, is the square root of the variance. It provides a measure of spread that is easier to interpret because it is in the same units as the original data.

• High variance: Data points are spread out over a wide range.
• Low variance: Data points are closer to the mean.
• Standard deviation is often used in reporting clinical data.

In our exercise, understanding variance and standard deviation helps evaluate how much blood pressure readings fluctuate, assisting in determining the effect of treatments.