6. Normal Distribution and Continuous Random Variables
Standard Normal Distribution
3:41 minutesProblem 5.1.23
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=0.33
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 = 0.33. The standard normal curve is a bell-shaped curve with a mean of 0 and a standard deviation of 1.
Step 2: Recall that the area under the standard normal curve represents probabilities. To find the area to the left of z = 0.33, you need to use the cumulative distribution function (CDF) for the standard normal distribution.
Step 3: Use the z-score table (also called the standard normal table) or technology (such as a graphing calculator, statistical software, or an online tool) to find the cumulative probability corresponding to z = 0.33. The table or tool will provide the area under the curve to the left of this z-score.
Step 4: If using a z-score table, locate the row corresponding to the first two digits of the z-score (0.3) and the column corresponding to the second decimal place (0.03). The intersection of this row and column gives the cumulative probability.
Step 5: If using technology, input the z-score (0.33) into the appropriate function for the standard normal CDF. For example, in a calculator, you might use a function like 'normalcdf(-∞, 0.33)' or 'P(Z ≤ 0.33)'. The result will be the area to the left of z = 0.33.
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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, a z-score indicates how far and in what direction a data point deviates from the mean, which is essential for finding areas under the curve.
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Area Under the Curve
The area under the curve of a probability distribution represents the probability 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 this case, finding the area to the left of z=0.33 involves calculating the cumulative probability up to that z-score.
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