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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 defined by different combinations of mean and standard deviation.

Step by step solution

01

Understanding the Normal Distribution

The normal distribution, often referred to as the bell curve, is a probability distribution that is symmetric about the mean. It describes how the values of a variable are distributed, with most observations clustering around the central peak and probabilities tapering off equally in both directions.
02

Identifying Parameters that Define a Normal Distribution

A normal distribution is not uniquely defined by a single set of parameters. Instead, it is characterized by two parameters: the mean (\( \mu \)) and the standard deviation (\( \sigma \)). The mean determines the center of the distribution, while the standard deviation measures how spread out the values are.
03

Recognizing a Family of Distributions

Since the mean and standard deviation can take any real number values, there are unlimited combinations of these parameters that define different normal distributions. This variability allows for a 'family' of normal distributions, each with its own unique set of characteristics based on the specific values of the mean and standard deviation.

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Key Concepts

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

Probability Distribution
Probability distributions play an essential role in understanding how probabilities are assigned to different outcomes. In the realm of statistics, they show the possible values that a random variable can take, alongside how likely these values are to occur. The normal distribution is a specific type of probability distribution, characterized by its bell-shaped curve. It is significant because many real-world phenomena fit this distribution pattern, which means they can be modeled and predicated accurately.
  • The normal distribution is continuous, meaning it deals with data that can take any value within a range.
  • The area under the curve of a probability distribution represents probabilities, summing to 1 or 100%.
Understanding probability distributions can enhance your ability to interpret data patterns and make informed predictions.
Mean and Standard Deviation
The mean and standard deviation are crucial parameters defining a normal distribution. The mean, often symbolized by \( \mu \), represents the central point of the distribution. It is essentially the average of all data points. This central tendency helps in understanding where most values cluster.

The standard deviation, denoted by \( \sigma \), indicates how much the values deviate from the mean. It measures the extent to which the data points spread out from the average value:
  • A small standard deviation indicates that the data points are close to the mean, resulting in a narrow bell curve.
  • A large standard deviation suggests widespread data, leading to a flatter and broader curve.
In a nutshell, while the mean provides a measure of central location, the standard deviation offers insights into the distribution's variability.
Family of Distributions
The term "family of distributions" refers to the collection of normal distributions that can result from different combinations of means and standard deviations. Each normal distribution has its unique bell shape determined by these two parameters. With infinite possibilities for the mean \( \mu \) and the standard deviation \( \sigma \), there are endless variations of normal distributions possible.

Some core facts about this family include:
  • Changing the mean shifts the entire graph left or right, altering its center without affecting the shape.
  • Modifying the standard deviation expands or contracts the graph, affecting the spread but not its peak's location.
Recognizing the family nature of these distributions shows that they are not a singular entity but rather a dynamic set characterized by diverse scenarios, suitable for modeling different data sets and phenomena.

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Most popular questions from this chapter

The weights of canned hams processed at the Henline Ham Company follow the normal distribution, with a mean of 9.20 pounds and a standard deviation of 0.25 pounds. The label weight is given as 9.00 pounds. a. What proportion of the hams actually weigh less than the amount claimed on the label? b. The owner, Glen Henline, is considering two proposals to reduce the proportion of hams below label weight. He can increase the mean weight to 9.25 and leave the standard deviation the same, or he can leave the mean weight at 9.20 and reduce the standard deviation from 0.25 pounds to \(0.15 .\) Which change would you recommend?

Management at Gordon Electronics is considering adopting a bonus system to increase production. One suggestion is to pay a bonus on the highest 5 percent of production based on past experience. Past records indicate weekly production follows the normal distribution. The mean of this distribution is 4,000 units per week and the standard deviation is 60 units per week. If the bonus is paid on the upper 5 percent of production, the bonus will be paid on how many units or more?

A study of long distance phone calls made from the corporate offices of the Pepsi Bottling Group, Inc., in Somers, New York, showed the calls follow the normal distribution. The mean length of time per call was 4.2 minutes and the standard deviation was 0.60 minutes. a. What fraction of the calls last between 4.2 and 5 minutes? b. What fraction of the calls last more than 5 minutes? c. What fraction of the calls last between 5 and 6 minutes? d. What fraction of the calls last between 4 and 6 minutes? e. As part of her report to the president, the Director of Communications would like to report the length of the longest (in duration) 4 percent of the calls. What is this time?

Assume that the mean hourly cost to operate a commercial airplane follows the normal distribution with a mean \(\$ 2,100\) per hour and a standard deviation of \(\$ 250 .\) What is the operating cost for the lowest 3 percent of the airplanes?

The weights of cans of Monarch pears follow the normal distribution with a mean of 1,000 grams and a standard deviation of 50 grams. Calculate the percentage of the cans that weigh: a. Less than 860 grams. b. Between 1,055 and \(1 ; 100\) grams. c. Between 860 and 1,055 grams.
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