6. Normal Distribution and Continuous Random Variables
Standard Normal Distribution
2:27 minutesProblem 5.1.35
Textbook Question
Finding Area In Exercises 23–36, find the indicated area under the standard normal curve. If convenient, use technology to find the area.
To the left of z= -1.28 and to the right of z= 1.28
Verified step by step guidance1
Step 1: Understand the problem. You are tasked with finding the area under the standard normal curve in two regions: (1) to the left of z = -1.28 and (2) to the right of z = 1.28. The standard normal curve is symmetric, with a mean of 0 and a standard deviation of 1.
Step 2: Recall that the total area under the standard normal curve is 1. To find the area to the left of z = -1.28, you need to use the cumulative distribution function (CDF) for the standard normal distribution. The CDF gives the area to the left of a specified z-value.
Step 3: To find the area to the right of z = 1.28, use the property that the area to the right of a z-value is equal to 1 minus the CDF value for that z-value. Mathematically, this can be expressed as: \( \text{Area to the right of } z = 1.28 = 1 - \text{CDF}(z = 1.28) \).
Step 4: Use technology (such as a graphing calculator, statistical software, or online tools) to find the CDF values for z = -1.28 and z = 1.28. Alternatively, you can use a standard normal table to look up these values. The table provides the cumulative area to the left of a given z-value.
Step 5: Add the two areas together to find the total area under the curve for the specified regions. Specifically, \( \text{Total Area} = \text{Area to the left of } z = -1.28 + \text{Area to the right of } z = 1.28 \).
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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 z-score, 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-scores
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 essential for determining the area under the standard normal curve, as they help identify the position of a value relative to the overall distribution.
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Area Under the Curve
The area under the curve in 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 distribution tables or technology. In the context of the question, finding the area to the left of z = -1.28 and to the right of z = 1.28 involves calculating the cumulative probabilities associated with these z-scores.
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