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: Analyze the trend

Looking at the data, as the number of coin flips increases, the number of heads up also increases. Let's check the ratio of heads to flips for some points:

• At 10 flips, 7 heads: $\frac{7}{10} = 0.7$
• At 20 flips, 12 heads: $\frac{12}{20} = 0.6$
• At 30 flips, 18 heads: $\frac{18}{30} = 0.6$
• At 40 flips, 23 heads: $\frac{23}{40} \approx 0.575$
• At 50 flips, 30 heads: $\frac{30}{50} = 0.6$
• At 60 flips, 35 heads: $\frac{35}{60} \approx 0.583$
• At 70 flips, 38 heads: $\frac{38}{70} \approx 0.543$
• At 80 flips, 42 heads: $\frac{42}{80} = 0.525$
• At 90 flips, 45 heads: $\frac{45}{90} = 0.5$

The ratio is approaching 0.5 as the number of flips increases, but let's also look at the linear trend. Let's calculate the slope between two points, say (50, 30) and (90, 45). The slope $m = \frac{45 - 30}{90 - 50} = \frac{15}{40} = 0.375$? Wait, no, maybe better to see the pattern. From 50 to 60 (10 flips), heads go from 30 to 35 (5 more). 60 to 70: 35 to 38 (3 more). 70 to 80: 38 to 42 (4 more). 80 to 90: 42 to 45 (3 more). Wait, maybe a better way is to see the linear approximation. Let's take two points: (0,0) and (90,45). The slope is $\frac{45 - 0}{90 - 0} = 0.5$. So the line of best fit might be close to $y = 0.5x$, where $x$ is the number of flips and $y$ is the number of heads. For $x = 100$, $y = 0.5 \times 100 = 50$. But let's check the last few points. At 90 flips, 45 heads. From 90 to 100 is 10 more flips. Let's see the increase from 80 to 90: 42 to 45 (3 heads in 10 flips). From 70 to 80: 38 to 42 (4 heads). From 60 to 70: 35 to 38 (3 heads). From 50 to 60: 30 to 35 (5 heads). The average increase per 10 flips: (5 + 3 + 4 + 3 + 3)/5 = 18/5 = 3.6. So from 90 (45 heads) to 100 (10 more flips), we can estimate 45 + 3.6 ≈ 48.6, close to 50? Wait, but the slope from (0,0) to (90,45) is 0.5, so 0.5*100=50. Alternatively, looking at the data, as flips increase, the number of heads is roughly half the flips. At 50 flips, 30 (close to 25, but a bit more), at 90 flips, 45 (exactly half). So the line of best fit is likely around $y = 0.5x$. So for 100 flips, 50 heads. But wait, the options are 48,50,51,53. Let's check the trend again. From 80 to 90: 42 to 45 (3 heads in 10 flips). So from 90 to 100, maybe 3 more? 45 + 3 = 48? But at 50 flips, 30 (which is 0.6 of 50), 60 flips 35 (0.583), 70 flips 38 (0.543), 80 flips 42 (0.525), 90 flips 45 (0.5). So the trend is approaching 0.5. So for 100 flips, using the line of best fit (which is approaching y=0.5x), the number of heads should be around 50. But let's check the difference between the actual heads and 0.5x:

• 10 flips: 7 vs 5 (diff +2)
• 20 flips: 12 vs 10 (diff +2)
• 30 flips: 18 vs 15 (diff +3)
• 40 flips: 23 vs 20 (diff +3)
• 50 flips: 30 vs 25 (diff +5)
• 60 flips: 35 vs 30 (diff +5)
• 70 flips: 38 vs 35 (diff +3)
• 80 flips: 42 vs 40 (diff +2)
• 90 flips: 45 vs 45 (diff 0)

Ah, so at 90 flips, the number of heads is exactly 0.590. So the line of best fit is converging to y = 0.5x. So for 100 flips, y = 0.5100 = 50. The difference at 90 is 0, so it's likely 50.

Step2: Confirm the line of best fit

Since the data points are approaching a linear relationship where the number of heads is approximately half the number of flips (as seen at 90 flips, 45 heads is half of 90), the line of best fit should be close to \( y = 0.5x \). For \( x = 100 \), \( y = 0.5 \times 100 = 50 \).

Answer:

B. 50 (assuming the options are labeled, but the correct value is 50)