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Chapter 6: Problem 6

What are the parameters of the normal distribution?

Short Answer

Expert verified
The parameters of the normal distribution are the mean \( \mu \) and the standard deviation \( \sigma \). The mean locates the center of the graph and the standard deviation determines the height and width of the graph. These two parameters fully characterize a normal distribution.

Step by step solution

01

Understand the Normal Distribution

The normal distribution, also known as the Gaussian 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. In graph form, normal distribution will appear as a bell curve.
02

Identify the First Parameter - Mean

The mean, denoted as \( \mu \), is the first parameter of the normal distribution. It is the average value of the data set and it determines the location of the center of the graph. A change in mean shifts the entire graph left or right on the horizontal scale.
03

Identify the Second Parameter - Standard Deviation

The standard deviation, denoted as \( \sigma \), is the second parameter of a normal distribution. It is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range.

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Most popular questions from this chapter

According to the U.S. Employment and Training Administration, the average weekly unemployment benefit paid out in 2008 was \(\$ 297\) (http://www.ows.doleta.gov/unemploy/hb394.asp). Suppose that the current distribution of weekly unemployment benefits paid out is approximately normally distributed with a mean of \(\$ 297\) and a standard deviation of \(\$ 74.42\). Find the probability that a randomly selected American who is receiving unemployment benefits is receiving a more than \(\$ 400\) per week b. between \(\$ 200\) and \(\$ 340\) per week

For the standard normal distribution, find the area within one standard deviation of the mean-that is, the area between \(\mu-\sigma\) and \(\mu+\sigma\).

A psychologist has devised a stress test for dental patients sitting in the waiting rooms. According to this test, the stress scores (on a scale of 1 to 10 ) for patients waiting for root canal treatments are found to be approximately normally distributed with a mean of \(7.59\) and a standard deviation of \(.73 .\) a. What percentage of such patients have a stress score lower than \(6.0\) ? b. What is the probability that a randomly selected root canal patient sitting in the waiting room has a stress score between \(7.0\) and \(8.0\) ? c. The psychologist suggests that any patient with a stress score of \(9.0\) or higher should be given a sedative prior to treatment. What percentage of patients waiting for root canal treatments would need a sedative if this suggestion is accepted?

According to an Allstate/National Journal poll, \(39 \%\) of the U.S. adults polled said that it is extremely or very likely that "there will be a female president within \(10-15\) years" in the United States (USA Today, March 28,2012 ). Suppose that this percentage is true for the current population of U.S. adults. Find the probability that in a random sample of 800 U.S. adults, more than 330 would hold the foregoing belief.

Tommy Wait, a minor league baseball pitcher, is notorious for taking an excessive amount of time between pitches. In fact, his times between pitches are normally distributed with a mean of 36 seconds and a standard deviation of \(2.5\) seconds. What percentage of his times between pitches are a. longer than 39 seconds? b. between 29 and 34 seconds?
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