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Chapter 7: Problem 8

List the major characteristics of a normal probability distribution.

Short Answer

Expert verified
The major characteristics include a symmetric bell shape, mean=median=mode, inflection points, the empirical rule, and total area of 1.

Step by step solution

01

Understand the Normal Distribution

A normal probability distribution is a type of continuous probability distribution for a real-valued random variable. It is also known as the Gaussian distribution.
02

Identify the Shape

The shape of a normal distribution is a symmetric bell curve. This means that the distribution is symmetric around its mean, which makes it visually balanced.
03

Symmetry

In a normal distribution, the left side is a mirror image of the right side. This symmetry indicates that the distribution has no skewness.
04

Center Mean and Equal Median and Mode

The mean, median, and mode of a normal distribution are equal and located at the center of the distribution. Thus, the highest point on the normal curve is at the mean.
05

Inflection Points

A normal distribution has inflection points at one standard deviation away from the mean on either side. These points are where the distribution changes curvature.
06

The Empirical Rule

According to the empirical rule, about 68% of the data falls within one standard deviation of the mean, approximately 95% falls within two standard deviations, and about 99.7% lies within three standard deviations.
07

Total Area Equals One

The total area under the normal distribution curve sums to one, indicating that it represents a total probability of 1.
08

Uniqueness by Mean and Standard Deviation

A normal distribution is completely specified by its mean and standard deviation. Different combinations of mean and standard deviation create distinct normal distributions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Gaussian Distribution
The Gaussian Distribution, commonly referred to as the Normal Distribution, is a cornerstone in statistics and probability theory. Think of it as a foundational model for how data points tend to cluster around an average. Imagine a bell-shaped curve鈥攖his iconic shape is precisely what a Gaussian Distribution looks like. It鈥檚 symmetrical, meaning its left half is a mirror image of its right half. This symmetry means that data is evenly distributed on both sides of the mean.
  • It鈥檚 continuous, allowing any real number to occur within a set range of values.
  • The center, known as the mean, is where we see the peak of this distribution. Here, the mean, median, and mode all coincide.
  • Two parameters鈥攎ean and standard deviation鈥攗niquely define it.
These characteristics make the Gaussian Distribution incredibly useful in fields such as economics, biology, and social sciences, as they can model a large variety of phenomena effectively.
Empirical Rule
The Empirical Rule is a handy guideline for understanding how data spreads within a Gaussian Distribution. It's also sometimes called the 68-95-99.7 Rule based on the proportions of data lying within one, two, and three standard deviations from the mean respectively. Here's how it works:
  • Approximately 68% of the data falls within one standard deviation from the mean. This shows a strong clustering around the center.
  • About 95% of the data is within two standard deviations, giving a larger range of spread but still central.
  • An impressive 99.7% of data lies within three standard deviations, covering nearly all data points in the distribution.
This rule is particularly useful for quickly gauging the spread and variability of data, making it easier to identify outliers or unusual values. By applying the empirical rule, analysts can make informed predictions and inferences about future observations or phenomena.
Symmetry in Distributions
Symmetry is a fascinating property of the Normal Distribution. It means that one half of the distribution is a perfect reflection of the other half, centered around the mean. This symmetry gives the distribution its characteristic bell shape and provides some valuable insights
  • The lack of skewness means most of the data points are evenly distributed around the mean.
  • Statistical measures like mean, median, and mode are identical and located at the center.
  • This uniformity helps in simplifying complex analyses since the behavior is predictable regardless of the direction.
When a distribution is symmetric, it's much easier to predict and understand its behavior because changes and relationships are more consistent. If a dataset follows a symmetric distribution, tools like the Empirical Rule and z-scores are much more effective, making statistical analysis more straightforward and informative.

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Most popular questions from this chapter

A recent article in the Cincinnati Enquirer reported that the mean labor cost to repair a heat pump is \(\$ 90\) with a standard deviation of \(\$ 22 .\) Monte's Plumbing and Heating Service completed repairs on two heat pumps this morning. The labor cost for the first was \(\$ 75\) and it was \(\$ 100\) for the second. Compute \(z\) values for each and comment on your findings.

Customers experiencing technical difficulty with their Internet cable hookup may call an 800 number for technical support. It takes the technician between 30 seconds to 10 minutes to resolve the problem. The distribution of this support time follows the uniform distribution. a. What are the values for \(a\) and \(b\) in minutes? b. What is the mean time to resolve the problem? What is the standard deviation of the time? c. What percent of the problems take more than 5 minutes to resolve. d. Suppose we wish to find the middle 50 percent of the problem-solving times. What are the end points of these two times?

Fast Service Truck Lines uses the Ford Super Duty F-750 exclusively. Management made a study of the maintenance costs and determined the number of miles traveled during the year followed the normal distribution. The mean of the distribution was 60,000 miles and the standard deviation 2,000 miles. a. What percent of the Ford Super Duty F-750s logged 65,200 miles or more? b. What percent of the trucks logged more than 57,060 but less than 58,280 miles? c. What percent of the Fords traveled 62,000 miles or less during the year? d. Is it reasonable to conclude that any of the trucks were driven more than 70,000 miles? Explain.

The annual sales of romance novels follow the normal distribution. However, the mean and the standard deviation are unknown. Forty percent of the time sales are more than 470,000 , and 10 percent of the time sales are more than \(500,000 .\) What are the mean and the standard deviation?

The goal at U.S. airports handling international flights is to clear these flights within 45 minutes. Let's interpret this to mean that 95 percent of the flights are cleared in 45 minutes, so 5 percent of the flights take longer to clear. Let's also assume that the distribution is approximately normal. a. If the standard deviation of the time to clear an international flight is 5 minutes, what is the mean time to clear a flight? b. Suppose the standard deviation is 10 minutes, not the 5 minutes suggested in part a. What is the new mean? c. A customer has 30 minutes from the time her flight landed to catch her limousine. Assuming a standard deviation of 10 minutes, what is the likelihood that she will be cleared in time?
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