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

number of coin flips	0	10	20	30	40	50	60	70	80	90
number of heads up	0	7	12	18	23	30	35	38	42	45

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: Calculate the slope and y - intercept

We can use the least - squares regression method. Let \(x\) be the number of coin flips and \(y\) be the number of heads up. First, we calculate the means of \(x\) and \(y\).
\(\bar{x}=\frac{0 + 10+20+\cdots+90}{10}=45\)
\(\bar{y}=\frac{0 + 7+12+\cdots+45}{10}=25\)
The slope \(m\) of the line of best - fit is given by the formula \(m=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\)
\(\sum_{i = 1}^{10}(x_{i}-\bar{x})(y_{i}-\bar{y})=(0 - 45)(0 - 25)+(10 - 45)(7 - 25)+\cdots+(90 - 45)(45 - 25)\)
\(=45\times25+35\times18 + 25\times13+15\times7+5\times2+5\times5+15\times10+25\times13+35\times17+45\times20\)
\(=1125+630+325 + 105+10+25+150+325+595+900\)
\(=4190\)
\(\sum_{i = 1}^{10}(x_{i}-\bar{x})^{2}=(0 - 45)^{2}+(10 - 45)^{2}+\cdots+(90 - 45)^{2}\)
\(=45^{2}+35^{2}+25^{2}+15^{2}+5^{2}+5^{2}+15^{2}+25^{2}+35^{2}+45^{2}\)
\(=2025+1225+625+225+25+25+225+625+1225+2025\)
\(=8250\)
\(m=\frac{4190}{8250}\approx0.51\)
The y - intercept \(b\) is given by \(b=\bar{y}-m\bar{x}\), \(b = 25-0.51\times45=25 - 22.95 = 2.05\)
The equation of the line of best - fit is \(y=0.51x + 2.05\)

Step2: Predict for \(x = 100\)

Substitute \(x = 100\) into the equation \(y=0.51x+2.05\)
\(y=0.51\times100+2.05=51 + 2.05=53.05\approx53\)

Answer:

53