Textbook Question
In Exercises 21–26, find the indicated probability using the standard normal distribution. If convenient, use technology to find the probability.
P(z < 1.28)
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6. Normal Distribution and Continuous Random Variables
Standard Normal Distribution
1:21 minutesProblem 5.RE.8
Textbook Question
In Exercises 7–18, find the indicated area under the standard normal curve. If convenient, use technology to find the area.
To the left of z = -1.95
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 = -1.95. This represents the cumulative probability for a z-score of -1.95 in a standard normal distribution.
Step 2: Recall that the standard normal distribution is symmetric about the mean (z = 0), and the total area under the curve is 1. The area to the left of a given z-score represents the cumulative probability up to that z-score.
Step 3: Use the z-score table (also called the standard normal table) or technology (such as a graphing calculator or statistical software) to find the cumulative probability corresponding to z = -1.95. The table or software will provide the area to the left of this z-score.
Step 4: If using a z-score table, locate the row corresponding to -1.9 and the column corresponding to 0.05 (since -1.95 = -1.9 + 0.05). The intersection of this row and column gives the cumulative probability.
Step 5: If using technology, input the z-score of -1.95 into the appropriate function (e.g., 'normalcdf' on a calculator or a similar function in statistical software) to directly obtain the cumulative probability. This value represents the area under the curve to the left of z = -1.95.
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Video duration:
1mKey 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 used to describe how data points are distributed in a standardized way, allowing for comparison across different datasets. The area under the curve represents probabilities, with the total area equaling 1.
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Z-scores
A Z-score indicates how many standard deviations a data point is from the mean of a distribution. It is calculated by subtracting the mean from the data point and dividing by the standard deviation. In the context of the standard normal distribution, a Z-score of -1.95 means the value is 1.95 standard deviations below the mean.
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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-tables or technology, such as statistical software, to determine probabilities associated with specific Z-scores.
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