The amount of time (in hours per week) a student utilizes a...

The amount of time (in hours per week) a student utilizes a math-tutoring center roughly follows the normal distribution $y=0.7979 e^{-(x-5.4)^{2} / 0.5}, \quad 4 \leq x \leq 7$ where $x$ is the number of hours. (a) Use a graphing utility to graph the function. (b) From the graph in part (a), estimate the average number of hours per week a student uses the tutoring center.

The amount of time (in hours per week) a student utilizes a math-tutoring center roughly follows the normal distribution
$y=0.7979 e^{-(x-5.4)^{2} / 0.5}, \quad 4 \leq x \leq 7$
where $x$ is the number of hours.
(a) Use a graphing utility to graph the function.
(b) From the graph in part (a), estimate the average number of hours per week a student uses the tutoring center.

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Key Concepts

Normal Distribution

This is a continuous probability distribution that is symmetric about its mean, showing that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its bell-shape and is defined by the parameters of mean (center) and variance (spread).

Probability Density Function (PDF)

A function that describes the likelihood of a random variable taking on a given value. In the context of the normal distribution, the PDF provides the relative probability for different outcomes and is used to understand the distribution's shape and characteristics.

Mean of the Distribution

In a normal distribution, the mean is the central value where the function peaks and is the point of symmetry. It represents the average outcome and is inferred as the center of the bell curve in the graphical representation of the normal distribution.

Graphical Representation

Graphing the probability density function using a graphing utility allows one to visualize the shape of the distribution, identify key features such as the peak (which corresponds to the mean), and estimate various properties such as the average value or the spread of the data.

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