Textbook Question
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?
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6. Normal Distribution and Continuous Random Variables
Standard Normal Distribution
2:03 minutesProblem 5.CR.6
Textbook Question
In Exercises 6–11, find the indicated area under the standard normal curve. If convenient, use technology to find the area.
To the left of z = 0.72
Verified step by step guidance1
Step 1: Understand the problem. You are tasked with finding the area under the standard normal curve to the left of z = 0.72. This represents the cumulative probability for a z-score of 0.72 in a standard normal distribution.
Step 2: Recall that the standard normal distribution is symmetric about the mean (z = 0), with a mean of 0 and a standard deviation of 1. The area under the curve to the left of a given z-score represents the cumulative probability up to that z-score.
Step 3: Use a z-table or technology (such as a graphing calculator, statistical software, or an online tool) to find the cumulative probability corresponding to z = 0.72. In a z-table, locate the row for 0.7 and the column for 0.02, as these correspond to the z-score of 0.72.
Step 4: If using technology, input the z-score of 0.72 into the cumulative distribution function (CDF) for the standard normal distribution. For example, in a calculator, you might use the function normcdf(-∞, 0.72) or its equivalent.
Step 5: Interpret the result. The value obtained from the z-table or technology represents the proportion of the data under the standard normal curve to the left of z = 0.72. This is the solution to the problem.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Standard Normal Distribution
The standard normal distribution is a special normal distribution with a mean of 0 and a standard deviation of 1. It is represented by the variable 'z', which indicates how many standard deviations an element is from the mean. This distribution is crucial for calculating probabilities and areas under the curve, as it allows for the standardization of different normal distributions.
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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. In the context of the standard normal distribution, the z-score indicates the position of a value along the curve, which is essential for determining the area to the left or right of that score.
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Area Under the Curve
The area under the curve of a probability distribution represents the likelihood of a random variable falling within a particular range. For the standard normal distribution, this area can be found using z-scores and standard normal tables or technology. In this case, finding the area to the left of z = 0.72 involves calculating the cumulative probability up to that z-score, which reflects the proportion of data points below that value.
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