Question

5 a box containing 1,000 coins is shaken, and the coins are emptied onto a table. only the coins that land heads up are returned to the box, and then the process is repeated. the accompanying table shows the number of trials and the number of coins returned to the box after each trial.

write an exponential regression equation, rounding the calculated values to the nearest hundredth. use the equation to predict how many coins would be returned to the box after the eighth trial.

Explanation:

Step1: Input data into calculator

Using a graphing calculator, input the trial numbers as \(x\)-values (where trial \(0\) is \(x = 0\), trial \(1\) is \(x=1\), etc.) and the number of coins returned as \(y\)-values.

Step2: Perform exponential regression

On the calculator, select the exponential regression function (\(y = ab^{x}\)). After running the regression, we get \(a\approx1000.00\) and \(b\approx0.61\). So the exponential regression equation is \(y = 1000(0.61)^{x}\).

Step3: Predict for \(x = 8\)

Substitute \(x = 8\) into the equation \(y=1000(0.61)^{8}\).

$$y = 1000\times(0.61)^{8}$$

$$y=1000\times0.0217$$

$$y = 21.7\approx22$$

Answer:

The exponential regression equation is \(y = 1000(0.61)^{x}\). The number of coins returned after the eighth trial is approximately \(22\).