Textbook Question
In Exercises 21–26, find the indicated probability using the standard normal distribution. If convenient, use technology to find the probability.
P(0.42 < z < 3.15)
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6. Normal Distribution and Continuous Random Variables
Standard Normal Distribution
3:59 minutesProblem 5.RE.11
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 = -2.825
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 = -2.825. 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 a given z-score, you can use a z-table (standard normal table) or technology such as a graphing calculator or statistical software.
Step 3: If using a z-table, locate the row corresponding to the first two digits of the z-score (-2.8) and the column corresponding to the hundredths place (0.025). The intersection of this row and column gives the cumulative probability to the left of z = -2.825.
Step 4: If using technology, input the z-score (-2.825) into the cumulative distribution function (CDF) for the standard normal distribution. For example, in a graphing calculator, use the function normcdf(-∞, -2.825, 0, 1), where -∞ represents the lower bound, 0 is the mean, and 1 is the standard deviation.
Step 5: Interpret the result. The value obtained represents the proportion of the data under the standard normal curve that lies to the left of z = -2.825. This is the desired area.
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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 used to describe how data is distributed in a standardized way, allowing for comparison across different datasets. The z-score indicates how many standard deviations an element is from the mean, facilitating the calculation of probabilities and areas under the curve.
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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 of -2.825 indicates that the value is 2.825 standard deviations below the mean, which is essential for determining the area under the curve to the left of this z-score.
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Area Under the Curve
The area under the curve in a standard normal distribution represents the probability of a random variable falling within a certain range. For a given z-score, this area can be found using statistical tables or technology, such as calculators or software. In this case, finding the area to the left of z = -2.825 helps determine the likelihood of a value being less than this z-score, which is crucial for various statistical analyses.
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