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Chapter 7: Problem 7

Explain what is meant by this statement: "There is not just one normal probability distribution but a 'family' of them."

Short Answer

Expert verified
The statement refers to the infinite variations of normal distributions determined by different mean and standard deviation values.

Step by step solution

01

Understanding 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 often represented as a bell-shaped curve.
02

Parameters of Normal Distribution

The normal distribution is characterized by two parameters: the mean (\(\mu\)) and the standard deviation (\(\sigma\)). These parameters determine the shape and location of the distribution.
03

Mean Defines the Center

The mean (\(\mu\)) is the central point of the distribution. Different normal distributions can have different means, which shifts the center of the bell curve along the horizontal axis.
04

Standard Deviation Determines the Spread

The standard deviation (\(\sigma\)) affects the spread or width of the distribution. A larger \(\sigma\) causes the bell curve to be wider and flatter, while a smaller \(\sigma\) results in a narrower and taller curve.
05

Family of Normal Distributions Explained

Since the normal distribution can have any real mean and positive standard deviation, there are infinitely many normal distributions depending on these parameter values. This is why it is referred to as a 'family' of distributions.

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Key Concepts

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

Probability Distribution
A probability distribution provides a mathematical description of the likelihood of various outcomes in a random experiment. Put simply, it tells us how probabilities are distributed over the values of a random variable. One of the most common types of probability distribution is the normal distribution, often visualized as a bell-shaped curve.

In a normal distribution:
  • The overall shape is symmetric, with the highest point at the mean.
  • Most data points cluster around the mean.
  • It represents how data values spread out, tapering off equally towards both extremes.
Understanding probability distributions helps in grasping how likely certain outcomes are compared to others.
Mean
The mean is a key parameter in a normal distribution. It represents the average or the central value of the set of data. In the context of the normal distribution, the mean (\( \mu \)) is at the center of the bell curve. It divides the distribution into two perfectly symmetrical halves.

The mean tells us:
  • Where the center of our data lies.
  • About the expected value of the dataset.
Different normal distributions can have different means, which simply shifts the curve along the horizontal axis. A higher mean shifts the curve to the right, whereas a lower mean shifts it to the left.
Standard Deviation
Standard deviation (\( \sigma \)) is a measure of the amount of variation or dispersion in a set of values. When it comes to normal distribution:
  • A larger standard deviation means the data is more spread out, resulting in a wider and flatter bell curve.
  • A smaller standard deviation indicates that the data is clustered closer around the mean, creating a steeper and narrower curve.
Standard deviation is crucial in determining how much variability there is from the mean. It helps identify how much of the data falls within a certain range around the mean. Typically, about 68% of the data falls within one standard deviation from the mean in a normal distribution.
Symmetry
Symmetry in normal distribution means that the left and right sides of the bell curve are mirror images of each other. This is a defining feature of the normal distribution. Due to this symmetry:
  • The mean, median, and mode of the distribution are all equal and coincide at the center.
  • Probabilities are equally likely on both sides of the mean.
The symmetry helps standardize the manner in which we can make predictions about data. Because of this consistent pattern, normal distributions are a powerful tool in probability and statistics for making informed decisions.

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Most popular questions from this chapter

List the major characteristics of a normal probability distribution.

The annual commissions earned by sales representatives of Machine Products Inc., a manufacturer of light machinery, follow the normal probability distribution. The mean yearly amount earned is \(\$ 40,000\) and the standard deviation is \(\$ 5,000 .\) a. What percent of the sales representatives earn more than \(\$ 42,000\) per year? b. What percent of the sales representatives earn between \(\$ 32,000\) and \(\$ 42,000 ?\) c. What percent of the sales representatives earn between \(\$ 32,000\) and \(\$ 35,000 ?\) d. The sales manager wants to award the sales representatives who earn the largest commissions a bonus of \(\$ 1,000\). He can award a bonus to \(20 \%\) of the representatives. What is the cutoff point between those who earn a bonus and those who do not?

According to the Insurance Institute of America, a family of four spends between \(\$ 400\) and \(\$ 3,800\) per year on all types of insurance. Suppose the money spent is uniformly distributed between these amounts. a. What is the mean amount spent on insurance? b. What is the standard deviation of the amount spent? c. If we select a family at random, what is the probability they spend less than \(\$ 2,000\) per year on insurance? d. What is the probability a family spends more than \(\$ 3,000\) per year?

Suppose the Internal Revenue Service reported that the mean tax refund for the year 2017 was \(\$ 2,800 .\) Assume the standard deviation is \(\$ 450\) and that the amounts refunded follow a normal probability distribution. a. What percent of the refunds are more than \(\$ 3,100 ?\) b. What percent of the refunds are more than \(\$ 3,100\) but less than \(\$ 3,500 ?\) c. What percent of the refunds are more than \(\$ 2,250\) but less than \(\$ 3,500 ?\)

According to media research, the typical American listened to 195 hours of music in the last year. This is down from 290 hours 4 years earlier. Dick Trythal is a big country and western music fan. He listens to music while working around the house, reading, and riding in his truck. Assume the number of hours spent listening to music follows a normal probability distribution with a standard deviation of 8.5 hours. a. If Dick is in the top \(1 \%\) in terms of listening time, how many hours did he listen last year? b. Assume that the distribution of times 4 years earlier also follows the normal probability distribution with a standard deviation of 8.5 hours. How many hours did the \(1 \%\) who listen to the least music actually listen?
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