Question

1. the table below shows the results of amber spinning a spinner with 10 equal sections. if the spinner is spun 80 more times, which two predictions are the most reasonable?

color	green	yellow	blue	red

a. the color red was spun 3 times more than the color blue.
b. the color yellow was spun 9 times more than the color red.
c. the color green was spun 4 more times than the color blue.
d. the color green was spun 8 more times than the color yellow.
e. the color blue was spun 8 more times than the color red.

Answer:

To solve this, we analyze each option using the given frequencies (Green: 7, Yellow: 9, Blue: 3, Red: 1) and the additional 80 spins (we assume the relative frequencies remain similar, so we can use the original frequencies to predict proportions, but for the "more times" comparisons, we can check the original differences and see if they hold with more spins, or just check the original differences first as a start):

Analyzing Option A:

Red frequency = 1, Blue frequency = 3.
Difference: \( 1 - 3 = -2 \) (Red is less than Blue, so "Red was spun 3 times more than Blue" is false).

Analyzing Option B:

Yellow frequency = 9, Red frequency = 1.
Difference: \( 9 - 1 = 8 \) (Not 9, so "Yellow was spun 9 times more than Red" is false).

Analyzing Option C:

Green frequency = 7, Blue frequency = 3.
Difference: \( 7 - 3 = 4 \) (So "Green was spun 4 times more than Blue" matches the original difference. With more spins, this relative difference is likely to hold, so this is reasonable).

Analyzing Option D:

Green frequency = 7, Yellow frequency = 9.
Difference: \( 7 - 9 = -2 \) (Green is less than Yellow, so "Green was spun 8 more times than Yellow" is false).

Analyzing Option E:

Blue frequency = 3, Red frequency = 1.
Difference: \( 3 - 1 = 2 \) (Not 8, but wait—wait, the problem says "the spinner is spun 80 more times". Wait, maybe we misinterpreted. Wait, the original spins: let's check the total original spins? Wait, the table has Green:7, Yellow:9, Blue:3, Red:1—total original spins: \( 7 + 9 + 3 + 1 = 20 \). Now, spinning 80 more times (total 100 spins) would scale the frequencies by 5 (since \( 80 + 20 = 100 \), and \( 100 / 20 = 5 \)).

• Scaled Green: \( 7 \times 5 = 35 \)
• Scaled Yellow: \( 9 \times 5 = 45 \)
• Scaled Blue: \( 3 \times 5 = 15 \)
• Scaled Red: \( 1 \times 5 = 5 \)

Now check Option E: Blue (15) vs Red (5). Difference: \( 15 - 5 = 10 \)? Wait, no—wait, the original difference between Blue and Red is \( 3 - 1 = 2 \). Scaled by 5, it’s \( 2 \times 5 = 10 \)? But the option says "Blue was spun 8 more times than Red". Hmm, maybe the initial approach was wrong. Wait, maybe the question is about the original spins (before the 80 more) or the total? Wait, the problem says "if the spinner is spun 80 more times"—so total spins become original + 80. Let’s re-express:

Original frequencies (let’s call them \( f \)):
\( f_{\text{Green}} = 7 \), \( f_{\text{Yellow}} = 9 \), \( f_{\text{Blue}} = 3 \), \( f_{\text{Red}} = 1 \).

Total original spins: \( N = 7 + 9 + 3 + 1 = 20 \).

Probability of each color:
\( P_{\text{Green}} = \frac{7}{20} \), \( P_{\text{Yellow}} = \frac{9}{20} \), \( P_{\text{Blue}} = \frac{3}{20} \), \( P_{\text{Red}} = \frac{1}{20} \).

For 80 more spins, the expected number of times each color is spun is \( 80 \times P_{\text{color}} \).

• Expected Green: \( 80 \times \frac{7}{20} = 28 \)
• Expected Yellow: \( 80 \times \frac{9}{20} = 36 \)
• Expected Blue: \( 80 \times \frac{3}{20} = 12 \)
• Expected Red: \( 80 \times \frac{1}{20} = 4 \)

Now, total spins (original + 80):

• Total Green: \( 7 + 28 = 35 \)
• Total Yellow: \( 9 + 36 = 45 \)
• Total Blue: \( 3 + 12 = 15 \)
• Total Red: \( 1 + 4 = 5 \)

Now re-analyze each option with total spins:

• Option A: Red (5) vs Blue (15). \( 5 - 15 = -10 \) (Red is less, so false).
• Option B: Yellow (45) vs Red (5). \( 45 - 5 = 40 \) (Not 9, false).
• Option C: Green (35) vs Blue (15). \( 35 - 15 = 20 \)? Wait, no—wait, the option says "Green was spun 4 more times than Blue" (original difference was 4, and with scaling, the difference scales by 5: \( 4 \times 5 = 20 \)? Wait, maybe the question is about the additional 80 spins, not total. Let’s check the additional 80 spins:

• Additional Green: 28, Additional Blue: 12. \( 28 - 12 = 16 \)? No. Wait, maybe the problem is simpler—just check the original frequency differences, as the "80 more times" is a distractor, or the question is about the original spins (maybe a typo). Let’s go back to original frequencies:

Original:

• Green:7, Blue:3 → 7 - 3 = 4 (matches Option C: "Green was spun 4 more times than Blue").
• Blue:3, Red:1 → 3 - 1 = 2 (no, but Option E: "Blue was spun 8 more times than Red"—wait, 3 - 1 = 2, but if we spin 80 more times, the expected difference for Blue - Red in 80 spins is \( 80 \times (\frac{3}{20} - \frac{1}{20}) = 80 \times \frac{2}{20} = 8 \). Ah! That’s it.

So for the 80 additional spins:

• Expected Blue in 80 spins: \( 80 \times \frac{3}{20} = 12 \)
• Expected Red in 80 spins: \( 80 \times \frac{1}{20} = 4 \)
• Difference: \( 12 - 4 = 8 \) → matches Option E: "The color blue was spun 8 more times than the color red".

And Option C: Original Green (7) - Original Blue (3) = 4 → "Green was spun 4 more times than Blue" (this is true for the original spins, and the relative frequency means it’s reasonable for the additional spins too, but the key is the 80 more spins for Option E). Wait, but the question says "which two predictions are the most reasonable". Wait, we must have missed that. Let’s re-express:

For the 80 additional spins:

• Option C: Green - Blue in original: 7 - 3 = 4. In 80 spins, the expected difference is \( 80 \times (\frac{7}{20} - \frac{3}{20}) = 80 \times \frac{4}{20} = 16 \)? No, wait, no—if we consider the additional 80 spins, the number of times each color is spun in those 80 is based on probability. So:

• Probability Green: \( \frac{7}{20} \), so in 80 spins: \( 80 \times \frac{7}{20} = 28 \)
• Probability Blue: \( \frac{3}{20} \), so in 80 spins: \( 80 \times \frac{3}{20} = 12 \)
• Difference: \( 28 - 12 = 16 \) → not 4. So maybe the question is about the original spins (before the 80 more), but the problem says "if the spinner is spun 80 more times". This is confusing.

Wait, let’s check the original frequencies again:

• Green:7, Blue:3 → 7 - 3 = 4 (Option C: "Green was spun 4 more times than Blue"—true for original spins).
• Blue:3, Red:1 → 3 - 1 = 2 (original), but in 80 more spins, expected Blue: 12, expected Red: 4 → 12 - 4 = 8 (Option E: "Blue was spun 8 more times than Red"—true for the 80 additional spins).

So the two reasonable predictions are Option C and Option E.

Final Answer:

The two most reasonable predictions are:
C. The color green was spun 4 more times than the color blue.
E. The color blue was spun 8 more times than the color red.