Investment on Gold or stock market

Suppose that the percentage annual return you obtain when you invest a dollar in gold or the stock market is dependent on the general state of the national economy as indicated below. For example, the probability that the economy will be in “boom” state is 0.15. In this case, if you invest in the stock market your return is assumed to be 25%; on the other hand if you invest in gold when the economy is in a “boom” state your return will be minus 30%. Likewise for the other possible states of the economy. Note that the sum of the probabilities has to be 1–and is.

State of economy	Probability	Market Return	Gold Return
Boom	0.15	25%	(-30%)
Moderate Growth	0.35	20%	(-9%)
Week Growth	0.25	5%	35%
No Growth	0.25	(-14%)	50%

 

Based on the expected return, would you rather invest your money in the stock market or in gold? Why?

Keep in mind that your post must be made by 11:55PM EASTERN time on Wednesday during the week in which a discussion question is posed.  I will evaluate your responses to each of these questions using a 0 to 10 point scale, and your contribution to each of the Discussion Forums will count as 1.25 percent of the overall course grade for a total of 10 percent.

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Random variables can be either discrete or continuous. Discrete variables and their distributions were explained in Chapter 5. Recall that a discrete variable cannot assume all values between any two given values of the variables. On the other hand, a continuous variable can assume all values between any two given values of the variables. Examples of continuous variables are the heights of adult men, body temperatures of rats, and cholesterol levels of adults. Many continuous variables, such as the examples just mentioned, have distributions that are bell-shaped, and these are called approximately normally distributed variables. For example, if a researcher selects a random sample of 100 adult women, measures their heights, and constructs a histogram, the researcher gets a graph similar to the one shown in Figure 6–1(a). Now, if the researcher increases the sample size and decreases the width of the classes, the histograms will look like the ones shown in Figure 6–1(b) and (c). Finally, if it were possible to measure exactly the heights of all adult females in the United States and plot them, the histogram would approach what is called a normal distribution , shown in Figure 6–1(d). This distribution is also known as a bell curve or a Gaussian distribution, named for the German mathematician Carl Friedrich Gauss (1777–1855), who derived its equation.

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Figure 6–1

Histograms for the Distribution of Heights of Adult Women

 

Figure 6–2

Normal and Skewed Distributions

 

Objective 1

Identify distributions as symmetric or skewed.

No variable fits a normal distribution perfectly, since a normal distribution is a theoretical distribution. However, a normal distribution can be used to describe many variables, because the deviations from a normal distribution are very small. This concept will be explained further in Section 6–1.

When the data values are evenly distributed about the mean, a distribution is said to be a symmetric distribution. (A normal distribution is symmetric.) Figure 6–2(a) shows a symmetric distribution. When the majority of the data values fall to the left or right of the mean, the distribution is said to be skewed. When the majority of the data values fall to the right of the mean, the distribution is said to be a negatively or left-skewed distribution. The mean is to the left of the median, and the mean and the median are to the left of the mode. See Figure 6–2(b). When the majority of the data values fall to the left of the mean, a distribution is said to be a positively or right-skewed distribution. The mean falls to the right of the median, and both the mean and the median fall to the right of the mode. See Figure 6–2(c).

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The “tail” of the curve indicates the direction of skewness (right is positive, left is negative). These distributions can be compared with the ones shown in Figure 3–1 in Chapter 3. Both types follow the same principles.

This chapter will present the properties of a normal distribution and discuss its applications. Then a very important fact about a normal distribution called the central limit theorem will be explained. Finally, the chapter will explain how a normal distribution curve can be used as an approximation to other distributions, such as the binomial distribution. Since a binomial distribution is a discrete distribution, a correction for continuity may be employed when a normal distribution is used for its approximation.

Objective 2

Identify the properties of a normal distribution.

6–1Normal Distributions

In mathematics, curves can be represented by equations. For example, the equation of the circle shown in Figure 6–3 is x2 + y2 = r2, where r is the radius. A circle can be used to represent many physical objects, such as a wheel or a gear. Even though it is not possible to manufacture a wheel that is perfectly round, the equation and the properties of a circle can be used to study many aspects of the wheel, such as area, velocity, and acceleration. In a similar manner, the theoretical curve, called a normal distribution curve, can be used to study many variables that are not perfectly normally distributed but are nevertheless approximately normal.

Figure 6–3

Graph of a Circle and an Application

 

The mathematical equation for a normal distribution is

 

where

e ≈ 2.718 (≈ means “is approximately equal to”)

π ≈ 3.14

μ = population mean

σ = population standard deviation

This equation may look formidable, but in applied statistics, tables or technology is used for specific problems instead of the equation.

Another important consideration in applied statistics is that the area under a normal distribution curve is used more often than the values on the y axis. Therefore, when a normal distribution is pictured, the y axis is sometimes omitted.

Circles can be different sizes, depending on their diameters (or radii), and can be used to represent wheels of different sizes. Likewise, normal curves have different shapes and can be used to represent different variables.

The shape and position of a normal distribution curve depend on two parameters, the mean and the standard deviation. Each normally distributed variable has its own normal distribution curve, which depends on the values of the variable’s mean and standard deviation. Figure 6–4(a) shows two normal distributions with the same mean values but different standard deviations. The larger the standard deviation, the more dispersed, or spread out, the distribution is. Figure 6–4(b) shows two normal distributions with the same standard deviation but with different means. These curves have the same shapes but are located at different positions on the x axis. Figure 6–4(c) shows two normal distributions with different means and different standard deviations.

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Figure 6–4

Shapes of Normal Distributions

 

Historical Note

The discovery of the equation for a normal distribution can be traced to three mathematicians. In 1733, the French mathematician Abraham DeMoivre derived an equation for a normal distribution based on the random variation of the number of heads appearing when a large number of coins were tossed. Not realizing any connection with the naturally occurring variables, he showed this formula to only a few friends. About 100 years later, two mathematicians, Pierre Laplace in France and Carl Gauss in Germany, derived the equation of the normal curve independently and without any knowledge of DeMoivre’s work. In 1924, Karl Pearson found that DeMoivre had discovered the formula before Laplace or Gauss.

A normal distribution is a continuous, symmetric, bell-shaped distribution of a variable.

The properties of a normal distribution, including those mentioned in the definition, are explained next.

Summary of the Properties of the Theoretical Normal Distribution

1.A normal distribution curve is bell-shaped.

2.The mean, median, and mode are equal and are located at the center of the distribution.

3.A normal distribution curve is unimodal (i.e., it has only one mode).

4.The curve is symmetric about the mean, which is equivalent to saying that its shape is the same on both sides of a vertical line passing through the center.

5.The curve is continuous; that is, there are no gaps or holes. For each value of X, there is a corresponding value of Y.

6.The curve never touches the x axis. Theoretically, no matter how far in either direction the curve extends, it never meets the x axis—but it gets increasingly closer.

7.The total area under a normal distribution curve is equal to 1.00, or 100%. This fact may seem unusual, since the curve never touches the x axis, but one can prove it mathematically by using calculus. (The proof is beyond the scope of this textbook.)

8.The area under the part of a normal curve that lies within 1 standard deviation of the mean is approximately 0.68, or 68%; within 2 standard deviations, about 0.95, or 95%; and within 3 standard deviations, about 0.997, or 99.7%. See Figure 6–5, which also shows the area in each region.

The values given in item 8 of the summary follow the empirical rule for data given in Section 3–2.

You must know these properties in order to solve problems involving distributions that are approximately normal.

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Figure 6–5

Areas Under a Normal Distribution Curve

 

Objective 3

Find the area under the standard normal distribution, given various z values.

The Standard Normal Distribution

Since each normally distributed variable has its own mean and standard deviation, as stated earlier, the shape and location of these curves will vary. In practical applications, then, you would have to have a table of areas under the curve for each variable. To simplify this situation, statisticians use what is called the standard normal distribution .

The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.

The standard normal distribution is shown in Figure 6–6.

The values under the curve indicate the proportion of area in each section. For example, the area between the mean and 1 standard deviation above or below the mean is about 0.3413, or 34.13%.

The formula for the standard normal distribution is

 

All normally distributed variables can be transformed into the standard normally distributed variable by using the formula for the standard score:

 

This is the same formula used in Section 3–3. The use of this formula will be explained in Section 6–3.

As stated earlier, the area under a normal distribution curve is used to solve practical application problems, such as finding the percentage of adult women whose height is between 5 feet 4 inches and 5 feet 7 inches, or finding the probability that a new battery will last longer than 4 years. Hence, the major emphasis of this section will be to show the procedure for finding the area under the standard normal distribution curve for any z value. The applications will be shown in Section 6–2. Once the X values are transformed by using the preceding formula, they are called z values. The z value is actually the number of standard deviations that a particular X value is away from the mean. Table E in Appendix C gives the area (to four decimal places) under the standard normal curve for any z value from –3.49 to 3.49.

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Figure 6–6

Standard Normal Distribution

 

Interesting Fact

Bell-shaped distributions occurred quite often in early coin-tossing and die-rolling experiments.

Finding Areas Under the Standard Normal Distribution Curve

For the solution of problems using the standard normal distribution, a four-step procedure is recommended with the use of the Procedure Table shown.

Step 1Draw the normal distribution curve and shade the area.

Step 2Find the appropriate figure in the Procedure Table and follow the directions given.

There are three basic types of problems, and all three are summarized in the Procedure Table. Note that this table is presented as an aid in understanding how to use the standard normal distribution table and in visualizing the problems. After learning the procedures, you should not find it necessary to refer to the Procedure Table for every problem.

Procedure Table

Finding the Area Under the Standard Normal Distribution Curve

1.To the left of any z value:

Look up the z value in the table and use the area given.

 

2.To the right of any z value:

Look up the z value and subtract the area from 1.

 

3.Between any two z values: