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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
The mean is 0 and the standard deviation is 1 in a standard normal distribution.

Step by step solution

01

Understanding the Standard Normal Distribution

A standard normal distribution is a special type of normal distribution. It has been standardized so that the mean, which is the average value, is 0. The distribution is symmetric about the mean.
02

Identifying the Characteristics of the Distribution

In any normal distribution, the mean is the measure of the center. For a standard normal distribution, this mean is always set to 0. This central point is important as it divides the curve into two identical halves.
03

Standard Deviation in Standard Normal Distribution

Standard deviation measures the dispersion of data from the mean. In a standard normal distribution, the standard deviation is set to 1. This means that data near the mean are within one unit of standard distance from it.

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Key Concepts

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

Understanding Mean in Statistics
The mean in statistics is essentially the average of a set of numbers. It is used to find the central point of the data set. To calculate the mean:
  • Add together all the numbers in a set.
  • Divide the sum by the total number of values.
The mean is a fundamental concept because it offers a quick snapshot of where most values in a distribution are centered. In the context of a standard normal distribution, the mean is set to 0. This standardization helps when comparing different distributions, as it allows everyone to start from the same central point. This mean also signifies the peak of the bell curve, where the data distribution is symmetrical.
The Role of Standard Deviation
Standard deviation is a measure of how spread out the numbers are in a data set. It's a key concept for understanding variation within a distribution. The formula for standard deviation looks complex, but an easy way to understand it is:
  • Find the mean of the data set.
  • Subtract the mean from each data point and square the result.
  • Calculate the average of those squared differences.
  • Take the square root of that average.
For a standard normal distribution, the standard deviation is fixed at 1. This indicates that the typical distance between any data point and the mean is one unit. Consequently, over 68% of the data in a standard normal distribution lies within one standard deviation of the mean, which is 0 in this case.
Exploring Normal Distribution
Normal distribution is often referred to as a bell curve due to its bell-like shape. It is a continuous probability distribution that is symmetrical around its mean. A few key points about normal distribution include:
  • The mean, median, and mode of the distribution are all equal.
  • Most values cluster around the central peak and probabilities for values further away from the mean taper off symmetrically on both sides.
  • About 68% of the data falls within one standard deviation from the mean, 95% within two, and 99.7% within three. This is known as the Empirical Rule.
In the realm of statistics, understanding normal distribution is crucial because it's the basis for many statistical tests and methods. It allows statisticians and researchers to make inferences and predict probabilities with confidence, given its predictably structured nature.

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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?

Assume that \(x\) has a normal distribution with the specified mean and standard deviation. Find the indicated probabilities. $$ P(3 \leq x \leq 6) ; \mu=4 ; \sigma=2 $$

Let \(z\) be a random variable with a standard normal distribution. Find the indicated probability, and shade the corresponding area under the standard normal curve. $$ P(z \geq 2.17) $$

Suppose \(5 \%\) of the area under the standard normal curve lies to the right of \(z\). Is \(z\) positive or negative?

Assuming that the heights of college women are normally distributed with mean 65 inches and standard deviation \(2.5\) inches (based on information from Statistical Abstract of the United States, 112 th Edition), answer the following questions. (Hint: Use Problems 5 and 6 and Figure 6-3.) (a) What percentage of women are taller than 65 inches? (b) What percentage of women are shorter than 65 inches? (c) What percentage of women are between \(62.5\) inches and \(67.5\) inches? (d) What percentage of women are between 60 inches and 70 inches?
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