George and Sandra both decide to open small cafes in town, on the same block: "George's Gourmet Tea" and "Sandra's Super Coffee." They soon become concerned about their effects on each other. Will having two beverage shops so close make them compete for a limited number of customers, or will it make their block a more popular destination and so attract more customers? Let a (t) represent the daily profit (scaled to have an appropriate unit, like hundreds of dollars) at George's cafe at time t, and let y(t) represent the daily profit at Sandra's cafe at time t. Note that a and y can take values that are positive (profit), negative (loss), and zero (break-even). We will model the rates of change of their daily profits with the linear system da -2л — Зу dt (3) dy -3x – 2y. dt (4) (a) Interpret the terms of the system and how they relate to the rates of change of their daily profits. In other words, what does this model assume is true about the way daily profit at one cafe affects its own and the other's daily profit rate of change? Explain thoroughly. (b) Solve the IVP comprised of the system above and the initial condition (x(0), y(0)) = (1, 0.75). Show your work and/or explain your process. (c) Describe the long-term predictions that the model makes for æ and y. Interpret these predictions in the context of George and Sandra's cafes.

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Inverse Normal Distribution

The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.

Mean, Median, Mode

It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.

Z-Scores

A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.