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Chapter 3: Problem 4

The graph of a Gaussian model is ________ shaped, where the ________ ________ is the maximum -value of the graph.

Short Answer

Expert verified
The graph of a Gaussian model is bell shaped, where the maximum y-value is the mean value of the function.

Step by step solution

01

Describe Gaussian Model

A Gaussian model, or a normal distribution, is a graph which represents data that clusters around a mean or average value. It depicts a statistical distribution of values around this mean.
02

Identify Shape

The graph of Gaussian model is bell shaped. It resembles the shape of a bell, which starts and ends near the x-axis and has a peak, making this shape symmetrical.
03

Determine Maximum Value

In a Gaussian model, the maximum y-value of the graph can be found at the peak of the 'bell'. This peak denotes the mean or average of the data set.

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Key Concepts

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

Normal Distribution
When we talk about a normal distribution, we're referring to a way of presenting data that's, quite literally, normal! It's also known as a Gaussian distribution. This type of distribution is key in statistics because it often naturally arises in various datasets. Think of it as a snapshot of a population where most of the values are close to the average, with fewer occurrences as you move away from the center. This pattern shows us how data tends to clump around a central value. There are some key properties of the normal distribution:
  • It is symmetric about the mean.
  • The mean, median, and mode all fall at the center.
  • Its tails taper off as you move away from the mean, indicating fewer extreme values in the data set.
Understanding the normal distribution helps statisticians and scientists make predictions about data tendencies. It's fundamental in probability theory and used to model real-world phenomena like heights, test scores, and measurement errors.
Bell Shaped Curve
The term 'bell shaped curve' is often used to describe the graphical form of a Gaussian distribution. This term gives a hint about what the shape looks like. Imagine the elegant contours of a bell standing on its rim. The highest point in the middle curves down smoothly, tapering equally on both sides. Why is this shape important? Well, it visually shows how data points are spread. Most points cluster around the highest point鈥攖he mean鈥攁nd as you move away from this peak, data points gradually and symmetrically decrease. The features of a bell curve include:
  • The peak of the curve represents the mean or the most common value in the dataset.
  • It symmetrically tapers off to both sides, representing outliers or less common values.
  • One standard deviation covers the majority of data points close to the mean.
This curve is so iconic because it repeats this common pattern across various types of data in natural, social, and tech sciences.
Mean or Average Value
The mean, or average, is a central concept in understanding the normal distribution and the bell-shaped curve. In statistics, the mean is the sum of all data values divided by the number of values. It's the point where the balance occurs in the data set. For a normal distribution, the mean has a special significance. It's the point where the peak of the bell curve resides. It's both the midpoint of symmetry and the center, dividing the distribution evenly in half. Some key aspects of the mean in relation to a normal distribution:
  • In any normally distributed dataset, the mean equals the median and mode.
  • The peak of the curve, the highest point, aligns exactly with the mean.
  • The mean helps to determine how the data is spread around the center point.
Understanding the mean allows you to quickly grasp the central tendency of data, revealing what a typical value might be. It's crucial for interpreting many statistical measures and making data-driven decisions.

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Most popular questions from this chapter

The values \(y\) (in billions of dollars) of U.S. currency in circulation in the years \(\begin{array}{lllll}2000 & \text { through } 2007 & \text { can be } & \text { modeled } & \text { by }\end{array}\) \(y=-451+444 \ln t, 10 \leq t \leq 17,\) where \(t\) represents the year, with \(t=10\) corresponding to 2000 . During which year did the value of U.S. currency in circulation exceed \$690 billion? (Source: Board of Governors of the Federal Reserve System)

A cup of water at an initial temperature of \(78^{\circ} \mathrm{C}\) is placed in a room at a constant temperature of \(21^{\circ} \mathrm{C}\). The temperature of the water is measured every 5 minutes during a half-hour period. The results are recorded as ordered pairs of the form \((t, T),\) where \(t\) is the time (in minutes) and \(T\) is the temperature (in degrees Celsius). \(\left(0,78.0^{\circ}\right),\left(5,66.0^{\circ}\right),\left(10,57.5^{\circ}\right),\left(15,51.2^{\circ}\right)\) \(\left(20,46.3^{\circ}\right),\left(25,42.4^{\circ}\right),\left(30,39.6^{\circ}\right)\) (a) The graph of the model for the data should be asymptotic with the graph of the temperature of the room. Subtract the room temperature from each of the temperatures in the ordered pairs. Use a graphing utility to plot the data points \((t, T)\) and \((t, T-21)\). (b) An exponential model for the data \((t, T-21)\) is given by \(T-21=54.4(0.964)^{t} .\) Solve for \(T\) and graph the model. Compare the result with the plot of the original data. (c) Take the natural logarithms of the revised temperatures. Use a graphing utility to plot the points \((t, \ln (T-21))\) and observe that the points appear to be linear. Use the regression feature of the graphing utility to fit a line to these data. This resulting line has the form \(\ln (T-21)=a t+b\). Solve for \(T,\) and verify that the result is equivalent to the model in part (b). (d) Fit a rational model to the data. Take the reciprocals of the \(y\) -coordinates of the revised data points to generate the points \(\left(t, \frac{1}{T-21}\right)\) Use a graphing utility to graph these points and observe that they appear to be linear. Use the regression feature of a graphing utility to fit a line to these data. The resulting line has the form \(\frac{1}{T-21}=a t+b\) Solve for \(T,\) and use a graphing utility to graph the rational function and the original data points. (e) Why did taking the logarithms of the temperatures lead to a linear scatter plot? Why did taking the reciprocals of the temperatures lead to a linear scatter plot?

The inverse function of the exponential function given by \(f(x)=a^{x}\) is called the _____ function with base \(a\).

Automobiles are designed with crumple zones that help protect their occupants in crashes. The crumple zones allow the occupants to move short distances when the automobiles come to abrupt stops. The greater the distance moved, the fewer g's the crash victims experience. (One \(g\) is equal to the acceleration due to gravity. For very short periods of time, humans have withstood as much as 40 g's.) In crash tests with vehicles moving at 90 kilometers per hour, analysts measured the numbers of g's experienced during deceleration by crash dummies that were permitted to move \(x\) meters during impact. The data are shown in the table. A model for the data is given by \(y=-3.00+11.88 \ln x+(36.94 / x),\) where \(y\) is the number of g's. $$ \begin{array}{|c|c|} \hline x & \text { g's } \\ \hline 0.2 & 158 \\ 0.4 & 80 \\ 0.6 & 53 \\ 0.8 & 40 \\ 1.0 & 32 \\ \hline \end{array} $$ (a) Complete the table using the model. $$ \begin{array}{|l|l|l|l|l|l|} \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 \\ \hline y & & & & & \\ \hline \end{array} $$ (b) Use a graphing utility to graph the data points and the model in the same viewing window. How do they compare? (c) Use the model to estimate the distance traveled during impact if the passenger deceleration must not exceed \(30 \mathrm{~g}\) 's. (d) Do you think it is practical to lower the number of g's experienced during impact to fewer than \(23 ?\) Explain your reasoning.

The projected populations of California for the years 2015 through 2030 can be modeled by \(P=34.696 e^{0.0098 t},\) where \(P\) is the population (in millions) and \(t\) is the time (in years), with \(t=15\) corresponding to \(2015 .\) (Source: U.S. Census Bureau) (a) Use a graphing utility to graph the function for the years 2015 through 2030 . (b) Use the table feature of a graphing utility to create a table of values for the same time period as in part (a). (c) According to the model, when will the population of California exceed 50 million?
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