Ch. 6 - Normal Probability Distributions

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook

Triola 14th Edition

Ch. 6 - Normal Probability Distributions

Problem 6.5.21b

Chapter 6, Problem 6.5.21b

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.

b. If each height is converted from inches to centimeters, are the heights in centimeters also normally distributed?

Verified step by step guidance

Understand the problem: The question asks whether converting a normally distributed dataset (heights in inches) to another unit (centimeters) preserves the normal distribution. This involves understanding the properties of normal distributions and transformations.

Recall the concept of linear transformations: A linear transformation of a random variable involves operations such as scaling (multiplying by a constant) and shifting (adding or subtracting a constant). In this case, converting inches to centimeters involves multiplying each value by a constant (2.54).

State the property of normal distributions under linear transformations: A key property of normal distributions is that if a random variable X is normally distributed, then any linear transformation of X, such as Y = aX + b (where a and b are constants), will also be normally distributed.

Apply the property to the given problem: Since the conversion from inches to centimeters is a linear transformation (Y = 2.54X, where X is the height in inches and Y is the height in centimeters), the resulting dataset in centimeters will also follow a normal distribution.

Conclude: The heights in centimeters are also normally distributed because the transformation from inches to centimeters is linear, and linear transformations preserve the normality of a 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 and the standard deviation. Many natural phenomena, including human heights, tend to follow this distribution.

Linear Transformations

A linear transformation involves changing a dataset by applying a linear function, such as scaling or shifting. When converting heights from inches to centimeters, a linear transformation is applied where each height is multiplied by a constant factor (2.54). This transformation preserves the shape of the distribution, meaning if the original data is normally distributed, the transformed data will also be normally distributed.

Statistical Invariance

Statistical invariance refers to the property that certain statistical characteristics remain unchanged under specific transformations. In the context of normal distributions, this means that if a dataset is normally distributed, applying linear transformations (like converting units) will not alter its normality. Thus, the heights in centimeters will also follow a normal distribution if the original heights in inches are normally distributed.

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