6. Normal Distribution and Continuous Random Variables
Standard Normal Distribution
3:07 minutesProblem 6.5.21a
Textbook Question
Transformations The heights (in inches) of women listed in Data Set 1 “Body Data†in Appendix B have a distribution that is approximately normal, so it appears that those heights are from a normally distributed population.
a. If 2 inches is added to each height, are the new heights also normally distributed?
Verified step by step guidance1
Understand the concept of a normal distribution: A normal distribution is a symmetric, bell-shaped curve where most of the data points cluster around the mean, and the probabilities taper off as you move further from the mean.
Recognize the effect of adding a constant to a dataset: Adding a constant value (e.g., 2 inches) to every data point in a dataset shifts the entire distribution by that constant amount without changing its shape.
Recall that the shape of a normal distribution is determined by its mean and standard deviation. Adding a constant affects the mean (it increases by the constant) but does not affect the standard deviation or the overall shape of the distribution.
Conclude that if the original dataset is normally distributed, adding a constant to each value will result in a new dataset that is also normally distributed, as the shape of the distribution remains unchanged.
Verify the reasoning by considering the properties of transformations: Shifting a dataset by a constant is a linear transformation, which preserves the normality of the distribution.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Normal Distribution
A normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its bell-shaped curve and is defined by two parameters: the mean (average) and the standard deviation (spread). Many natural phenomena, including human heights, tend to follow this distribution.
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Transformation of Data
Transforming data involves applying a mathematical operation to each data point in a dataset. In this case, adding a constant (2 inches) to each height shifts the entire distribution without altering its shape. This means that while the mean height increases, the overall distribution remains normal, preserving its properties.
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Properties of Normal Distribution under Transformation
One key property of normal distributions is that they remain normal under linear transformations, which include adding or subtracting a constant. This means that if you add a constant to a normally distributed variable, the result is still normally distributed, with the mean adjusted by the constant but the standard deviation unchanged.
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