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Chapter 6: Problem 4

What are the values of the mean and standard deviation of a standard normal distribution?

Short Answer

Expert verified
Mean is 0 and standard deviation is 1.

Step by step solution

01

Understand the Standard Normal Distribution

The standard normal distribution is a special type of normal distribution with specific characteristics. It is often used as a reference point in statistics.
02

Define the Parameters of a Standard Normal Distribution

In any normal distribution, the mean is represented by \( \mu \) (mu) and the standard deviation by \( \sigma \) (sigma). For the standard normal distribution, these parameters have specific values.
03

Identify the Mean of the Standard Normal Distribution

The mean of a standard normal distribution is \( \mu = 0 \). This means that the distribution is centered around zero, making it symmetric about this point.
04

Identify the Standard Deviation of the Standard Normal Distribution

The standard deviation of a standard normal distribution is \( \sigma = 1 \). This indicates that the spread or dispersion of values in this distribution is measured using this unit distance.

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Key Concepts

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

Mean
In the realm of statistics, the mean is a crucial concept, often serving as a measure of central tendency. It's essentially the average value of a data set.
When we talk about the mean in the context of a normal distribution, we refer to it with the Greek letter \( \mu \). This value indicates where the center of the distribution lies.
In a standard normal distribution, this mean is standardized to be \( \mu = 0 \). This zero-centered distribution provides a reference point for comparing other distributions.
This unique characteristic of having a mean of zero helps simplify many statistical calculations and allows for a straightforward application of probability theories.
Standard Deviation
The standard deviation is another key measurement in statistics, representing the degree of variation or dispersion of a set of values.
In a normal distribution, it's symbolized by \( \sigma \), which defines how spread out the data points are around the mean.
For a standard normal distribution, the standard deviation is fixed at \( \sigma = 1 \). This standard unit measurement implies that the distribution has been scaled to reflect a uniform spread of data.
This characteristic is crucial for converting any normal distribution into a standard normal distribution through a process called standardization, thereby facilitating comparison.
Normal Distribution Characteristics
Understanding the normal distribution is fundamental in statistics as it describes a common pattern for the distribution of data.
A normal distribution is often likened to the classic 'bell curve', symmetric and featuring a single, peak value at the mean.
The standard normal distribution is a specific form of this, with a mean of \( 0 \) and a standard deviation of \( 1 \), effectively making it the benchmark or reference distribution.
  • Simplicity: The values around the mean decrease symmetrically, forming the bell-shaped curve.
  • Symmetry: The distribution is equally balanced around its mean.
  • Empirical Rule: Approximately 68% of data falls within one standard deviation from the mean in a normal distribution, expanding to about 95% within two standard deviations.
These characteristics make the normal distribution essential for probability calculations and the understanding of statistical phenomena.

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Most popular questions from this chapter

Police response time to an emergency call is the difference between the time the call is first received by the dispatcher and the time a patrol car radios that it has arrived at the scene (based on information from The Denver Post). Over a long period of time, it has been determined that the police response time has a normal distribution with a mean of \(8.4\) minutes and a standard deviation of \(1.7\) minutes. For a randomly received emergency call, what is the probability that the response time will be (a) between 5 and 10 minutes? (b) less than 5 minutes? (c) more than 10 minutes?

Do you try to pad an insurance claim to cover your deductible? About \(40 \%\) of all U.S. adults will try to pad their insurance claims! (Source: Are You Normal?, by Bernice Kanner, St. Martin's Press.) Suppose that you are the director of an insurance adjustment office. Your office has just received 128 insurance claims to be processed in the next few days. What is the probability that (a) half or more of the claims have been padded? (b) fewer than 45 of the claims have been padded? (c) from 40 to 64 of the claims have been padded? (d) more than 80 of the claims have not been padded?

Consider a binomial experiment with 20 trials and probability \(0.45\) of success on a single trial. (a) Use the binomial distribution to find the probability of exactly 10 successes. (b) Use the normal distribution to approximate the probability of exactly 10 successes. (c) Compare the results of parts (a) and (b).

A person's blood glucose level and diabetes are closely related. Let \(x\) be a random variable measured in milligrams of glucose per deciliter (1/10 of a liter) of blood. After a 12 -hour fast, the random variable \(x\) will have a distribution that is approximately normal with mean \(\mu=85\) and standard deviation \(\sigma=25\) (Source: Diagnostic Tests with Nursing Implications, edited by \(S .\) Loeb, Springhouse Press). Note: After 50 years of age, both the mean and standard deviation tend to increase. What is the probability that, for an adult (under 50 years old) after a 12 -hour fast, (a) \(x\) is more than 60 ? (b) \(x\) is less than \(110 ?\) (c) \(x\) is between 60 and 110? (d) \(x\) is greater than 140 (borderline diabetes starts at 140\() ?\)

Describe how the variability of the \(\bar{x}\) distribution changes as the sample size increases.
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