6. Normal Distribution and Continuous Random Variables
Standard Normal Distribution
0:51 minutesProblem 5.RE.21
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)
Verified step by step guidance1
Step 1: Understand the problem. The question asks for the probability that the standard normal variable z is less than 1.28, denoted as P(z < 1.28). The standard normal distribution has a mean of 0 and a standard deviation of 1.
Step 2: Recall that the cumulative distribution function (CDF) of the standard normal distribution gives the probability that z is less than or equal to a given value. In this case, we are looking for the CDF value at z = 1.28.
Step 3: Use a standard normal distribution table (z-table) or technology (such as a calculator or statistical software) to find the cumulative probability corresponding to z = 1.28. Locate the row and column in the z-table that correspond to 1.28, or input the value into a software tool.
Step 4: Interpret the result from the z-table or technology. The value obtained represents the area under the standard normal curve to the left of z = 1.28, which is the probability P(z < 1.28).
Step 5: If using technology, ensure the input is correct (e.g., using a function like norm.cdf(1.28) in Python or a similar function in other software). Verify the result matches the expected value from the z-table for accuracy.
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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 variable Z, which allows for the calculation of probabilities and percentiles for any normal distribution by standardizing values. This distribution is symmetric and bell-shaped, making it a fundamental concept in statistics for understanding how data is distributed.
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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. A Z-score indicates how many standard deviations an element is from the mean, allowing for comparison across different datasets and facilitating the use of the standard normal distribution for probability calculations.
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Cumulative Probability
Cumulative probability refers to the probability that a random variable takes on a value less than or equal to a specific value. In the context of the standard normal distribution, it is represented by the area under the curve to the left of a given Z-score. This concept is crucial for finding probabilities associated with specific Z-scores, such as P(z < 1.28), which can be determined using standard normal distribution tables or technology.
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