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Chapter 6: Problem 5

Briefly describe the standard normal distribution curve.

Short Answer

Expert verified
The standard normal distribution is a distribution where the mean is 0 and the standard deviation is 1, often represented as \(N(0,1)\). The curve is bell-shaped and symmetrical around the mean which is the highest point. Approximately 68% of values lie within one standard deviation, 95% within two standard deviations, and 99.7% within three standard deviations. The area under the curve equals 1 and represents the total probability space, while the probabilities for a variable are given by the area under the curve to the left of the variable.

Step by step solution

01

Definition

The Standard Normal Distribution is a special type of normal distribution where the mean (\(\mu\)) is equal to 0 and the standard deviation (\(\sigma\)) is equal to 1. Notationally this can be represented as \(N(0,1)\). This distribution is also referred to as a \(Z\) distribution.
02

Shape Description

The shape of a standard normal distribution curve is a bell-shape. It is perfectly symmetrical about its mean which is 0. The highest point on the curve is at the mean. Also, approximately 68% of the data falls within one standard deviation of the mean (between -1 and 1 in this case), 95% within two standard deviations, and 99.7% within three standard deviations. The tails of the curve extend to infinity without touching the horizontal axis.
03

Understanding Properties

There are a few important properties of the standard normal distribution: 1) It is symmetric around its mean, 2) The area under the curve represents the total probability space and equals to 1, 3) It has a peak at the mean, 4) The probabilities for the variable are given by the area under the curve to the left of \(Z\)

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Most popular questions from this chapter

Find the area under the standard normal curve a. between \(z=0\) and \(z=4.28\) b. from \(z=0\) to \(z=-3.75\) \(c\). to the right of \(z=7.43\) d. to the left of \(z=-4.69\)

Suppose you are conducting a binomial experiment that has 15 trials and the probability of success of \(.02\). According to the sample size requirements, you cannot use the normal distribution to approximate the binomial distribution in this situation. Use the mean and standard deviation of this binomial distribution and the empirical rule to explain why there is a problem in this situation. (Note: Drawing the graph and marking the values that correspond to the empirical rule is a good way to start.)

According to the 2007 American Time Use Survey by the Bureau of Labor Statistics, employed adults living in households with no children younger than 18 years engaged in leisure activities for \(4.4\) hours a day on average (Source: http://www.bls.gov/news.release/atus,nr0.htm). Assume that currently such times are (approximately) normally distributed with a mean of \(4.4\) hours per day and a standard deviation of \(1.08\) hours per day. Find the probability that the amount of time spent on leisure activities per day for a randomly chosen individual from the population of interest (employed adults living in households with no children younger than 18 years) is a. between \(3.0\) and \(5.0\) hours per day b. less than \(2.0\) hours per day

Let \(x\) be a continuous random variable that is normally distributed with a mean of 25 and a standard deviation of 6 . Find the probability that \(x\) assumes a value a. between 29 and 36 b. between 22 and 35

One of the cars sold by Walt's car dealership is a very popular subcompact car called Rhino. The final sale price of the basic model of this car varies from customer to customer depending on the negotiating skills and persistence of the customer. Assume that these sale prices of this car are normally distributed with a mean of \(\$ 19,800\) and a standard deviation of \(\$ 350\). a. Dolores paid \(\$ 19,445\) for her Rhino. What percentage of Walt's customers paid less than Dolores for a Rhino? b. Cuthbert paid \(\$ 20,300\) for a Rhino. What percentage of Walt's customers paid more than Cuthbert for a Rhino?
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