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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

\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|}
\hline
\begin{tabular}{l}
Number of \\ Coin Flips
\end{tabular} & 0 & 10 & 20 & 30 & 40 & 50 & 60 & 70 & 80 & 90 \\
\hline
\begin{tabular}{l}
Number of \\ Heads Up
\end{tabular} & 0 & 7 & 12 & 18 & 23 & 30 & 35 & 38 & 42 & 45 \\
\hline
\end{tabular}

According to the line of best fit, about how many times would the coin land heads up in 100 flips?

A. 48
B. 50
C. 51
D. 53



Answer :

GinnyAnswer
To determine how many times the coin would land heads up in 100 flips, we need to find the line of best fit based on the given data.

The line of best fit can be expressed in the form [tex]\( y = mx + b \)[/tex], where:
- [tex]\( y \)[/tex] is the number of heads up,
- [tex]\( x \)[/tex] is the number of coin flips,
- [tex]\( m \)[/tex] is the slope of the line,
- [tex]\( b \)[/tex] is the y-intercept of the line.

From the data provided:
```
Number of Coin Flips (x): 0, 10, 20, 30, 40, 50, 60, 70, 80, 90
Number of Heads Up (y): 0, 7, 12, 18, 23, 30, 35, 38, 42, 45
```

Based on a detailed analysis, the slope [tex]\( m \)[/tex] (rate at which heads up increases per flip) and y-intercept [tex]\( b \)[/tex] (the number of heads up when flips are zero) is calculated as:
- Slope ([tex]\( m \)[/tex]) = 0.5078787878787878
- Y-intercept ([tex]\( b \)[/tex]) = 2.145454545454546

Using this equation of the line of best fit:
[tex]\[ y = 0.5078787878787878 \cdot x + 2.145454545454546 \][/tex]

We substitute [tex]\( x = 100 \)[/tex] (for 100 coin flips) into the equation to find the predicted number of heads up:
[tex]\[ y = 0.5078787878787878 \cdot 100 + 2.145454545454546 \][/tex]
[tex]\[ y = 50.78787878787878 + 2.145454545454546 \][/tex]
[tex]\[ y \approx 52.93333333333333 \][/tex]

Thus, according to the line of best fit, the coin would land heads up about 53 times in 100 flips. Therefore, the correct answer is:
53

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