Finding Specified Data Values In Exercises 31–38, answer the questions about the specified normal distribution.


Red Blood Cell Count The red blood cell counts (in millions of cells per microliter) for a population of adult males can be approximated by a normal distribution, with a mean of 5.4 million cells per microliter and a standard deviation of 0.4 million cells per microliter.


a. What is the minimum red blood cell count that can be in the top 25% of counts?

Verified step by step guidance

1. Step 1: Understand the problem. We are tasked with finding the minimum red blood cell count that falls in the top 25% of a normal distribution. This means we need to find the value (let's call it X) such that 25% of the data lies above it. This corresponds to finding the 75th percentile of the distribution.
2. Step 2: Recall the properties of a normal distribution. The given distribution has a mean (μ) of 5.4 million cells per microliter and a standard deviation (σ) of 0.4 million cells per microliter. The formula to standardize a value (convert it to a z-score) is: $Z=\frac{X-\mu }{\sigma }$.
3. Step 3: Use the z-score table or a statistical calculator to find the z-score corresponding to the 75th percentile. The cumulative probability up to this z-score is 0.75. From standard z-tables, the z-score for the 75th percentile is approximately $Z=0.674$.
4. Step 4: Rearrange the z-score formula to solve for X (the red blood cell count): $X=\mu +Z\times \sigma$. Substitute the values: $X=5.4+0.674\times 0.4$.
5. Step 5: Perform the calculation to find the value of X. This will give the minimum red blood cell count that falls in the top 25% of the distribution. Ensure the units are consistent (millions of cells per microliter).

Key Concepts

Normal Distribution

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, defined by its mean and standard deviation. In this context, the red blood cell counts follow a normal distribution, which allows us to use statistical methods to determine probabilities and percentiles.

Mean and Standard Deviation

The mean is the average of a set of values, while the standard deviation measures the amount of variation or dispersion in a set of values. In the given problem, the mean red blood cell count is 5.4 million cells per microliter, and the standard deviation is 0.4 million. These parameters are essential for calculating the specific data values and understanding the distribution of red blood cell counts.

Percentiles

A percentile is a measure used in statistics indicating the value below which a given percentage of observations fall. For example, the top 25% of red blood cell counts corresponds to the 75th percentile. To find this value in a normal distribution, one can use the mean and standard deviation along with z-scores, which represent the number of standard deviations a data point is from the mean.