Question

violet creates two spinners for a game. each spinner is spun once, and the sum is recorded. the table represents the sums of the spinners and the frequency of each sum.

possible sums

sum	frequency
7	2
9	3
11	4
13	3
15	2
17	1

what statement is true about the mean of the sums of the two spinners?

• the mean is 12.
• the mean is 16.
• the mean is the same as the median.
• the mean is the same as the range.

Explanation:

Step1: Calculate Total Frequency

Sum the frequencies: \(1 + 2 + 3 + 4 + 3 + 2 + 1 = 16\).

Step2: Calculate Sum of (Sum × Frequency)

• \(5×1 = 5\)
• \(7×2 = 14\)
• \(9×3 = 27\)
• \(11×4 = 44\)
• \(13×3 = 39\)
• \(15×2 = 30\)
• \(17×1 = 17\)

Total sum: \(5 + 14 + 27 + 44 + 39 + 30 + 17 = 176\).

Step3: Calculate Mean

Mean = \(\frac{176}{16} = 11\)? Wait, no, wait: Wait, 5+14=19, 19+27=46, 46+44=90, 90+39=129, 129+30=159, 159+17=176. 176/16=11? Wait, no, that can't be. Wait, wait, the frequencies: 1+2=3, +3=6, +4=10, +3=13, +2=15, +1=16. Correct. Then sum of products: 51=5, 72=14 (total 19), 93=27 (46), 114=44 (90), 133=39 (129), 152=30 (159), 17*1=17 (176). 176/16=11? Wait, but let's check median. The data is ordered: 5 (1), 7 (2), 9 (3), 11 (4), 13 (3), 15 (2), 17 (1). The total number of data points is 16, so median is average of 8th and 9th terms. Let's list the positions:

1:5, 2-3:7, 4-6:9, 7-10:11, 11-13:13, 14-15:15, 16:17.

8th term: 11, 9th term:11. So median is (11+11)/2=11. Wait, so mean is 11? But the options: Wait, maybe I miscalculated. Wait, no, wait the sum of products: 51=5, 72=14, 93=27, 114=44, 133=39, 152=30, 171=17. Let's add again: 5+14=19, +27=46, +44=90, +39=129, +30=159, +17=176. 176 divided by 16: 1611=176. So mean is 11. Median: 8th and 9th terms. Let's count the cumulative frequency:

• Sum 5: cumulative 1
• Sum 7: cumulative 1+2=3
• Sum 9: cumulative 3+3=6
• Sum 11: cumulative 6+4=10
• Sum 13: cumulative 10+3=13
• Sum 15: cumulative 13+2=15
• Sum 17: cumulative 15+1=16

So the 8th term is in the 11 group (since cumulative up to 9 is 6, up to 11 is 10). So 7th term is 11 (position 7), 8th term 11, 9th term 11. So median is 11. So mean (11) equals median (11). So the correct option is "The mean is the same as the median".

Answer:

The mean is the same as the median.