91Ó°ÊÓ

The distribution of white blood cell count per cubic millimeter of whole blood is approximately Normal with mean 7500 and standard deviation 1750 for healthy patients. Use technology or a table to answer these questions. For each include an appropriately labeled and shaded Normal curve. a. What is the probability that a randomly selected person will have a white blood cell count between 6000 and 10,000 ? b. An elevated white blood cell count can be a sign of infection somewhere in the body. A white blood cell count can be considered elevated if it is over 10,500 . What percentage of people have white blood cell counts in this elevated range? c. A white blood cell count below 4500 is considered low. People in this range may be referred for additional medical testing. What is the probability that a randomly selected person has a white blood cell count below 4500 ?

Step by step solution

01

Calculate Z-Scores

The z-score for each boundary value will be calculated. The formula to calculate a z-score is given by: \(Z = \frac{X - \mu}{\sigma}\), where \(X\) is a score, \(\mu\) is the mean and \(\sigma\) is the standard deviation. It's important to remember that the z-score tells us how many standard deviations away from the mean a data point is. For question (a), the z-scores for 6000 and 10000 are: \(Z_1 = \frac{6000 - 7500}{1750} = -0.86\) and \(Z_2 = \frac{10000 - 7500}{1750} = 1.43\). Similarly, z-scores for questions (b) and (c) can be calculated.

02

Use Standard Normal Table

Next, we use a standard normal distribution table (or technology, if you prefer) to find the probabilities that correspond to the calculated z-scores.

03

Find probabilities

The probabilities for the z-scores that we calculated in step 1 (rounded to two decimal places) are: Probability for -0.86 = 0.1949, and Probability for 1.43 = 0.9236. To find the probability of the blood count falling between 6000 and 10000 for question (a), we subtract the smaller probability from the larger one: \(P(6000 < X < 10000) = P(Z_2) - P(Z_1) = 0.9236 - 0.1949 = 0.7287\). The same procedure is repeated for questions (b) and (c) to find the probabilities.

04

Interpret Results

Finally, the results are interpreted correctly: for question (a), the probability that a randomly selected person will have a white blood cell count between 6000 and 10,000 is 0.7287 or 72.87%. For question (b), this would be the percentage of people whose cell count is above 10500. And for question (c), it is the probability that a randomly selected person will have a cell count below 4500.

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Key Concepts

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

White Blood Cell Count

White Blood Cell Count (WBC) is an essential health indicator often used in medical diagnostics. The Normal Distribution is frequently employed to analyze WBC due to its predictable pattern in the general population.

White blood cell counts are measured in cells per cubic millimeter of blood. For healthy individuals, a normal range is typically around 4,500 to 10,500 cells. This range can vary slightly depending on medical advice. A higher WBC can indicate infection, stress, or inflammation, while a lower count may suggest bone marrow issues or autoimmune diseases.

In statistical problems such as the one in this exercise, we model this count as a Normal Distribution with a specific mean and standard deviation. This allows for evaluating probabilities and values, providing insight into how rare or common a specific WBC is.

Z-Score Calculation

The Z-Score Calculation is vital in assessing whether a particular WBC is within the expected range of a healthy population. It helps in normalizing a data point within the context of a distribution.

The formula for a z-score is: \[Z = \frac{X - \mu}{\sigma}\]where:

• \(X\) is the observed value
• \(\mu\) is the mean of the distribution
• \(\sigma\) represents the standard deviation

A z-score displays the number of standard deviations a measure is from the mean. For example, a z-score of 1.43 means the value is 1.43 standard deviations above the average.

The benefit of this calculation lies in its ability to translate diverse numerical data onto a common scale. This makes it simpler to compare values across different distribution scenarios.

Probability Interpretation

Understanding Probability Interpretation helps in making meaning from z-scores. This concept involves analyzing what percentage or likelihood a particular score will occur given the Normal Distribution.

Probabilities are estimated by referring to a standard normal distribution table, which maps out the cumulative distribution function for z-values. For example, finding a probability that a WBC falls between 6000 and 10000 involves computing the area under the curve within those z-scores.

This interpretation is crucial in real-world applications where decisions are made based on statistical probabilities. Medical practitioners, for example, can assess health risks by understanding the probability of a patient having an abnormal WBC count.

Statistical Analysis

Statistical Analysis is the toolset enabling us to explore, interpret, and infer conclusions from data sets as diverse as white blood cell counts.

Using the concepts of Normal Distribution, Z-Scores, and Probability Interpretation, one can form clear and reliable judgments about the data. In this context, statistical analysis transforms mere numbers into actionable insights.

For instance, after determining the probabilities for varying WBCs, these conclusions help guide decision-making in the healthcare domain. Clinicians can suggest further testing or treatment plans based on whether a patient's counts fall into normal or abnormal categories.

• These methods not only support individual diagnosis but also help in understanding broader public health trends.
• Data-driven insights, derived from statistical analysis, promote evidence-based medicine.

Understanding these aspects of statistical analysis empowers students to delve into deeper investigations and recognize patterns within data commonly ignored without such a framework.