Textbook Question
In Exercises 7–18, find the indicated area under the standard normal curve. If convenient, use technology to find the area.
Between z = -1.55 and z = 1.04
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6. Normal Distribution and Continuous Random Variables
Standard Normal Distribution
1:42 minutesProblem 5.RE.2
Textbook Question
In Exercises 1 and 2, use the normal curve to estimate the mean and standard deviation.
Verified step by step guidance1
Step 1: Observe the normal curve provided in the image. The curve is symmetric, which indicates that the data follows a normal distribution. The peak of the curve represents the mean (μ), and the spread of the curve is determined by the standard deviation (σ).
Step 2: Identify the mean (μ) from the graph. The mean is located at the center of the curve, which corresponds to the value of 55 on the x-axis.
Step 3: Estimate the standard deviation (σ) using the red dashed lines. These lines mark one standard deviation away from the mean on both sides. From the graph, the values at one standard deviation are approximately 50 and 60.
Step 4: Calculate the standard deviation (σ) by finding the distance between the mean and one of the dashed lines. For example, σ = |55 - 50| = 5 or σ = |60 - 55| = 5.
Step 5: Summarize the findings: The mean (μ) is 55, and the standard deviation (σ) is 5. These values describe the normal distribution represented by the curve.
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Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Normal Distribution
The normal distribution is a continuous probability distribution characterized by its bell-shaped curve, symmetric about the mean. It is defined by two parameters: the mean (average) and the standard deviation (spread). In a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% within two, and 99.7% within three, making it essential for statistical analysis.
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Mean
The mean is the average value of a set of numbers, calculated by summing all values and dividing by the count of values. In the context of a normal distribution, the mean represents the center of the distribution, where the highest point of the curve occurs. It is a measure of central tendency that provides a useful summary of the data set.
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Standard Deviation
Standard deviation is a statistic that quantifies the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates a wider spread. In the context of the normal distribution, it helps determine the width of the curve and the proportion of data within specific ranges around the mean.
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