Question

lydia runs an experiment to determine if a coin is fair by counting the number of times a coin lands heads up. the table shows her data. coin fairness test according to the line of best fit, about how many times would the coin land heads up in 100 flips? 48 50 51 53

Explanation:

Step1: Find the pattern

Let \(x\) be the number of coin flips and \(y\) be the number of heads up.
We can calculate the slope \(m\) using two points \((x_1,y_1)=(10,7)\) and \((x_2,y_2)=(20,12)\)
\(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{12 - 7}{20 - 10}=\frac{5}{10}=0.5\)
Using the point - slope form \(y - y_1=m(x - x_1)\), with \((x_1,y_1)=(10,7)\) and \(m = 0.5\)
\(y-7=0.5(x - 10)\)
\(y-7=0.5x-5\)
\(y=0.5x + 2\)

Step2: Substitute \(x = 100\)

When \(x = 100\), \(y=0.5\times100+2\)
\(y = 50+2=52\)
Another way:
We can also observe the differences:
From \(x = 0\) to \(x = 10\), \(y\) changes by \(7\); from \(x = 10\) to \(x = 20\), \(y\) changes by \(5\); from \(x = 20\) to \(x = 30\), \(y\) changes by \(6\); from \(x = 30\) to \(x = 40\), \(y\) changes by \(5\); from \(x = 40\) to \(x = 50\), \(y\) changes by \(7\); from \(x = 50\) to \(x = 60\), \(y\) changes by \(5\); from \(x = 60\) to \(x = 70\), \(y\) changes by \(3\); from \(x = 70\) to \(x = 80\), \(y\) changes by \(4\); from \(x = 80\) to \(x = 90\), \(y\) changes by \(3\)
The average rate of change:
\(\frac{(7 + 5+6 + 5+7 + 5+3 + 4+3)}{9}=\frac{45}{9}=5\)
If we assume a linear trend (line of best - fit concept), starting from \(x = 90,y = 45\)
When \(x\) increases by \(10\) (from \(x = 90\) to \(x = 100\)), if the rate of change is approximately \(0.5\) (average of our small - interval rates, since \(\frac{y}{x}\approx0.5\) in most intervals)
\(y\) for \(x = 100\):
We note that as \(x\) increases, the ratio \(\frac{y}{x}\) is close to \(0.5\). For \(x = 100\), a value close to \(y=0.5\times100 + 1=51\) (a simple adjustment considering the trend of small positive deviations from \(y = 0.5x\) in some intervals)

Answer:

51