Question

Cerebral blood flow (CBF) in the brains of healthy people is normally distributed with a mean of 74 and a standard deviation of 16 a. What proportion of healthy people will have CBF readings between 60 and \(80 ?\) b. What proportion of healthy people will have CBF readings above \(100 ?\) c. If a person has a CBF reading below \(40,\) he is classified as at risk for a stroke. What proportion of healthy people will mistakenly be diagnosed as "at risk"?

Short answer

Answer: Approximately 45.54% of healthy people will have CBF readings between 60 and 80, 5.16% will have readings above 100, and 1.66% will have readings below 40.

Step-by-step solution

Understanding the given information and calculating z-scores for the intervals

We have the mean μ = 74 and the standard deviation σ = 16. We need to find the z-scores for the given CBF values in each part of the problem. The formula for calculating the z-score is: $$z = \frac{x - \mu}{\sigma}$$ where x is the CBF value, μ is the mean, and σ is the standard deviation.

Solving Part a - Proportion of people with CBF readings between 60 and 80

First, we need to find the z-scores for both 60 and 80 using the formula mentioned in step 1. For x=60: $$z_1 = \frac{60 - 74}{16} = -0.875$$ For x=80: $$z_2 = \frac{80 - 74}{16} = 0.375$$ Now we have to find the probability between these two z-scores. P(60 < CBF < 80) = P(-0.875 < Z < 0.375) We can look up these z-scores in the standard normal distribution table and find the probabilities. P(Z < -0.875) ≈ 0.1909 P(Z < 0.375) ≈ 0.6463 We subtract the two probabilities to find the proportion of people with CBF between 60 and 80. P(-0.875 < Z < 0.375) ≈ 0.6463 - 0.1909 = 0.4554 Approximately 45.54% of healthy people will have CBF readings between 60 and 80.

Solving Part b - Proportion of people with CBF readings above 100

First, find the z-score for x=100: $$z = \frac{100 - 74}{16} = 1.625$$ Now, we find the probability of having a CBF reading above 100. P(CBF > 100) = P(Z > 1.625) We look up the z-score in the standard normal distribution table: P(Z < 1.625) ≈ 0.9484 Since we want the probability above 1.625, we can calculate: P(Z > 1.625) = 1 - P(Z < 1.625) = 1 - 0.9484 = 0.0516 Approximately 5.16% of healthy people will have CBF readings above 100.

Solving Part c - Proportion of people with CBF readings below 40 and risk of stroke diagnosis

First, find the z-score for x=40: $$z = \frac{40 - 74}{16} = -2.125$$ Now, we find the probability of having a CBF reading below 40. P(CBF < 40) = P(Z < -2.125) We look up the z-score in the standard normal distribution table: P(Z < -2.125) ≈ 0.0166 Approximately 1.66% of healthy people will have CBF readings below 40. This means that 1.66% of healthy people will mistakenly be diagnosed as "at risk" for a stroke.

Key concepts

Understanding Z-Score

The z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is measured in terms of standard deviations from the mean. If you are given a z-score of 0, it means that the data point is exactly at the mean. A positive z-score indicates that the data point is above the mean, while a negative z-score signifies that it is below the mean.

For example, in the exercise about cerebral blood flow (CBF), z-scores are calculated to determine how a person's CBF reading compares to the average healthy individual's reading. A z-score is calculated using the formula:
\[z = \frac{x - \mu}{\sigma}\]
where \(x\) is the value of interest, \(\mu\) is the mean, and \(\sigma\) is the standard deviation of the dataset. In this way, z-scores allow comparison between different data points in a distribution, or between a data point and the mean of a distribution, providing insight on how 'unusual' a reading is.

The Role of Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the data points tend to be close to the mean, whereas a high standard deviation indicates that the data is spread out over a wider range of values.

Using our CBF example, the standard deviation is 16. This tells us that the CBF readings of healthy people typically vary by 16 units above or below the mean, which is 74. To put this into perspective, consider that a CBF value that is one standard deviation away from the mean (either above or below) would fall within the range most people are expected to be. Readings that are two or three standard deviations away from the mean are much rarer and could be considered outliers. The standard deviation is crucial when calculating z-scores, as it helps standardize different data points for comparison.

Probability in Normal Distribution

Probability, in the context of a normal distribution, refers to the likelihood of a certain range of outcomes occurring. The total area under a normal distribution curve represents a probability of 1 (or 100%). Probabilities for specific intervals are represented as areas under the curve between two points, often calculated using z-scores.

For instance, when we want to find the proportion of people with CBF readings between 60 and 80, we need to calculate the probability that a person’s CBF falls within this interval. This is done by finding the z-scores for 60 and 80 and then determining the area under the standard normal distribution curve between these z-scores. This area represents the probability we are looking for and provides insight into how common or rare that range of readings is among the healthy population.