6. Normal Distribution and Continuous Random Variables
Standard Normal Distribution
1:33 minutesProblem 5.1.42
Textbook Question
Finding Probability In Exercises 41–46, find the probability of z occurring in the shaded region of the standard normal distribution. If convenient, use technology to find the probability.
Verified step by step guidance1
Step 1: Understand the problem. The goal is to find the probability of z occurring in the shaded region of the standard normal distribution. The shaded region is between z = 0 and z = 1.96.
Step 2: Recall that the standard normal distribution has a mean of 0 and a standard deviation of 1. The z-scores represent the number of standard deviations away from the mean.
Step 3: Use the cumulative probability function for the standard normal distribution to find the area under the curve from z = 0 to z = 1.96. This can be done using statistical tables (z-tables) or technology such as a calculator or software.
Step 4: Look up the cumulative probability for z = 1.96 in the z-table or calculate it using technology. This value represents the area under the curve from z = -∞ to z = 1.96.
Step 5: Subtract the cumulative probability for z = 0 (which is 0.5, as it is the midpoint of the standard normal distribution) from the cumulative probability for z = 1.96 to find the probability of z occurring in the shaded region.
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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 case of the normal distribution where the mean is 0 and the standard deviation is 1. It is represented by the z-score, which indicates how many standard deviations an element is from the mean. The area under the curve represents probabilities, with the total area equaling 1.
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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, z-scores allow us to determine the probability of a value falling within a certain range.
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Probability and Area Under the Curve
In statistics, the probability of a z-score occurring within a certain range in the standard normal distribution is represented by the area under the curve for that range. This area can be calculated using z-tables or technology, and it reflects the likelihood of a random variable falling within that specified range.
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