Problem 18 - Thinking Mathematically

Question

Systolic blood pressure readings are normally distributed with a mean of 121 and a standard deviation of 15. (A reading above 140 is considered to be high blood pressure.) In Exercises 17-26, begin by converting any given blood pressure reading or readings into \(z\)-scores. Then use Table \(12.16\) on page 822 to find the percentage of people with blood pressure readings below \(148 .\)

Step-by-Step Solution

Answer

To find the percentage of individuals with blood pressure readings below 148, a z-score must first be calculated using the formula \(z = (x - \mu) / \sigma\), where \(x = 148\), \(\mu = 121\), and \(\sigma = 15\). The resulting z-score will be used with the standard normal distribution table to find the relevant percentage.

Step 1: Calculation of the Z-Score

The z-score, \(z\), for a value \(x\) in a distribution with mean \(\mu\) and standard deviation \(\sigma\), is calculated as: \(z = (x - \mu) / \sigma\). Here, \(x = 148\), \(\mu = 121\), and \(\sigma = 15\). Substituting these values will give us the z-score.

Step 2: Obtaining the Percentage

After calculating the z-score, it will be used with the standard normal distribution table (Table 12.16 on page 822) to find the percentage of individuals with blood pressure readings below 148. This percentage can be found by locating the z-score in the table and finding the corresponding percentage.

Key Concepts

Z-Score Calculation

The z-score is a very useful tool in statistics for determining how far away a data point is from the mean of a distribution. It's like a standard ruler that helps compare different values or observations in a consistent way. To calculate a z-score, you follow this simple formula:

• \( z = \frac{x - \mu}{\sigma} \)

Here, \(x\) represents the data point of interest. The symbol \(\mu\) is the mean of the population, and \(\sigma\) is the standard deviation.
Using this formula, the z-score can quantify how many standard deviations an element is from the mean. To relate this to the context of blood pressure: if you have a blood pressure reading, say 148, you subtract the mean blood pressure of 121 from it and then divide by the standard deviation of 15. This gives the z-score for the reading of 148. Understanding this score helps in comparing variations from typical, healthy blood pressure values.

Standard Deviation

Standard deviation is a measure of spread in a set of values. It's used to describe the amount of variation or dispersion in a dataset. In simpler terms, the standard deviation tells us how much the data varies from the average.

• If you have a small standard deviation, it means the data points are close to the mean.
• If you have a large standard deviation, the data points are spread over a wider range of values.

The calculation involves several steps, but it essentially measures the average difference between each data point and the mean. For blood pressure readings:

• If the standard deviation is 15, as in our example, most individual blood pressure readings lie within 15 units above or below the mean (121 in this case).
• A smaller standard deviation would indicate more consistency with the average, whereas a larger one would show greater variability among readings.

Understanding standard deviation is crucial for interpreting the significance of z-scores, because it affects how far typical values stray from the mean.

Blood Pressure Statistics

Blood pressure statistics are essential for understanding what constitutes a healthy reading and what may be a cause for concern.
Blood pressure is the force exerted by circulating blood against the walls of the arteries, and it's often measured in terms of systolic pressure, which is considered normally distributed.

• In the context of systolic blood pressure readings, a value of above 140 is classified as high blood pressure.
• A value below this suggests normal or healthy ranges, assuming an individual mean systolic reading of 121 with a standard deviation of 15.

These statistics allow health professionals to determine the health status of individuals by comparing specific readings to these benchmarks. By transforming a given reading to a z-score and checking it against known distributions, one can identify how typical or atypical a reading is. For example, by determining the percentage of people with readings below a specific level, such as 148, healthcare providers can gain insights into the health profile of a broader population.
Ultimately, these statistics help shape the understanding of cardiovascular health and guide decision-making in medical contexts.