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Chapter 4: Problem 63

Can the graph of a Gaussian model ever have an \(x\) -intercept? Explain.

Short Answer

Expert verified
No, a Gaussian model graph never has an \(x\)-intercept because it is asymptotic to the x-axis, meaning that it gets infinitely close but never touches or crosses the x-axis.

Step by step solution

01

Understanding Gaussian model graph

A graph of Gaussian distribution is a smoothly tapering, symmetrical bell shaped curve centered around a mean value \(\mu\). It has a peak at the mean, indicating the highest probability and it tapers off towards infinity on either sides. The function of the curve is given by the equation \(f(x) = \frac{1}{\sigma \sqrt{2\pi}}e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}\) where \(\sigma\) represents standard deviation.
02

Analyzing the properties of the graph

Since the equation has an exponential term, the range of \(f(x)\) is \(0 \leq f(x) < \infty\). This means the values it can take on the vertical \(y\)-axis can be zero or greater but never negative. Also, the curve is always centered around a mean value and symmetric about the vertical line \(x = \mu\).
03

Determining the x-intercepts

By definition, \(x\)-intercepts are the \(x\) values where \(f(x) = 0\). In the case of a Gaussian distribution, the curve tapers off towards \(\pm \infty\) on the x-axis but never touches or crosses it, making its \(y\) value asymptotic to zero. So it doesn't have any \(x\)-intercepts because it never crosses or touches the x-axis.

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Key Concepts

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

Understanding X-Intercepts in Gaussian Distribution
In mathematics, an \(x\)-intercept is where a graph crosses the \(x\)-axis. This means the function value at that point is zero. For the Gaussian distribution, the equation is \(f(x) = \frac{1}{\sigma \sqrt{2\pi}}e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}\). Due to the exponential factor, the graph starts at zero, rises to a peak at the mean, and then tapers off back towards zero. But crucially, it never actually touches the \(x\)-axis apart from at infinity.
  • This makes the \(f(x)\) value approach zero but never equals zero.
  • Thus, the Gaussian curve doesn't have any \(x\)-intercepts.
Bell Curve Shape of Gaussian Distribution
A Gaussian graph resembles a bell, hence the term "bell curve." It's symmetrical and smooth, peaking at the mean value. This symmetry means it's identical on both sides of the center.
  • The peak indicates the highest frequency or probability.
  • The tails taper off towards infinity, never really touching the axis.
This shape is significant because it shows the distribution of values around the mean, emphasizing that data near the mean are more frequent in occurrence.
Role of Standard Deviation in Gaussian Curve
Standard deviation, represented as \(\sigma\), measures the spread of values in a Gaussian distribution. It influences the width of the bell curve.
  • A larger \(\sigma\) results in a wider and flatter curve, indicating more spread in the data.
  • A smaller \(\sigma\) creates a narrower and taller curve, showing less variation around the mean.
The standard deviation is crucial for understanding how much variation there is from the mean, affecting how we interpret data spread.
Mean Value's Importance in Gaussian Distribution
In a Gaussian distribution, the mean value, denoted as \(\mu\), is the center of the bell curve. It represents the average or "expected" value.
  • The mean is the point of symmetry for the Gaussian curve.
  • All data is evenly distributed around this central point.
Understanding the mean provides insight into the central tendency of the data, guiding predictions and analyses.

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Most popular questions from this chapter

(a) use a graphing utility to graph the function, (b) find the domain, (c) use the graph to find the open intervals on which the function is increasing and decreasing, and (d) approximate any relative maximum or minimum values of the function. Round your results to three decimal places. $$f(x)=(\ln x)^{2}$$

Use the zero or root feature of a graphing utility to approximate the solution of the logarithmic equation. $$\ln x^{2}-e^{x}=-3-\ln x^{2}$$

Use a graphing utility to graph \(f\) and \(g\) in the same viewing window. Approximate the point of intersection of the graphs of \(f\) and \(g .\) Then solve the equation \(f(x)=g(x)\) algebraically. $$\begin{aligned}&f(x)=3 \log _{5} x\\\&g(x)=6\end{aligned}$$

The number \(Y\) of yeast organisms in a culture is given by the model $$Y=\frac{663}{1+72 e^{-0.547 t}}$$ where \(t\) represents the time (in hours). (a) Use a graphing utility to graph the model. (b) Use the model to predict the populations for the 19th hour and the 30 th hour. (c) According to this model, what is the limiting value of the population? (d) Why do you think this population of yeast follows a logistic growth model instead of an exponential growth model?

You take a five-pound package of steaks out of a freezer at 11 A.M. and place it in a refrigerator. Will the steaks be thawed in time to be grilled at 6 P.M.? Assume that the refrigerator temperature is \(40^{\circ} \mathrm{F}\) and the freezer temperature is \(0^{\circ} \mathrm{F}\). Use the formula (derived from Newton's Law of Cooling) $$t=-5.05 \ln \frac{T-40}{0-40}$$ where \(t\) is the time in hours (with \(t=0\) corresponding to 11 A.M.) and \(T\) is the temperature of the package of steaks (in degrees Fahrenheit).
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