Red Blood Cell Count Use the normal distribution in Exercise 16.


a. What percent of the adult males have a red blood cell count less than 6 million cells per microliter?

Verified step by step guidance

1. Step 1: Identify the parameters of the normal distribution. Typically, the problem will provide the mean (μ) and standard deviation (σ) for the red blood cell count distribution. If these values are not explicitly given, refer to Exercise 16.a for the necessary information.
2. Step 2: Standardize the value of 6 million cells per microliter using the z-score formula: $z=\frac{x-\mu }{\sigma }$, where x is the value of interest (6 million), μ is the mean, and σ is the standard deviation.
3. Step 3: Once the z-score is calculated, use a standard normal distribution table or a statistical software to find the cumulative probability corresponding to the z-score. This cumulative probability represents the proportion of adult males with a red blood cell count less than 6 million cells per microliter.
4. Step 4: Interpret the cumulative probability as a percentage. Multiply the cumulative probability by 100 to express the result as a percentage.
5. Step 5: Verify the result by ensuring the z-score calculation and lookup in the standard normal table are accurate. Double-check the parameters (mean and standard deviation) used in the calculations.

Key Concepts

Normal Distribution

The normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its bell-shaped curve and is defined by two parameters: the mean (average) and the standard deviation (which measures the spread of the data). In statistics, many natural phenomena, including biological measurements like red blood cell counts, tend to follow this distribution.

Z-Score

A Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is calculated by subtracting the mean from the value and then dividing by the standard deviation. Z-scores are useful for determining how many standard deviations an element is from the mean, allowing for comparisons across different datasets and helping to find probabilities in a normal distribution.

Percentile

A percentile is a measure used in statistics indicating the value below which a given percentage of observations in a group of observations falls. For example, the 50th percentile is the median, meaning that half of the data points are below this value. In the context of the red blood cell count, calculating the percentile helps determine the percentage of adult males with counts below a specific threshold, such as 6 million cells per microliter.