Question

the table compares the expected outcomes to the actual outcomes of the sums of 36 rolls of 2 standard number cubes. which comparisons are true of the frequency table? check all that apply. both data sets are symmetrical. the mean of the actual sums is more than the mean of the expected sums. the range of the sums is the same for both sets of data. the standard deviation of the actual sums is less than that of the expected sums. the medians and quartiles are the same in both sets.

Explanation:

Step1: Check symmetry

Expected data: \(1,2,3,4,5,6,5,4,3,2,1\) (symmetrical). Actual data: \(1,0,5,5,6,5,3,3,4,3,1\) (not symmetrical). So first option is false.

Step2: Calculate mean

Expected mean: \(\frac{(2\times1)+(3\times2)+(4\times3)+(5\times4)+(6\times5)+(7\times6)+(8\times5)+(9\times4)+(10\times3)+(11\times2)+(12\times1)}{36}=\frac{2 + 6+12 + 20+30+42+40+36+30+22+12}{36}=\frac{252}{36} = 7\)
Actual mean: \(\frac{(2\times1)+(3\times0)+(4\times5)+(5\times5)+(6\times6)+(7\times5)+(8\times3)+(9\times3)+(10\times4)+(11\times3)+(12\times1)}{36}=\frac{2+0 + 20+25+36+35+24+27+40+33+12}{36}=\frac{254}{36}\approx7.06\)
So second option is true.

Step3: Calculate range

Range = maximum - minimum. For both sets, maximum \(= 12\), minimum \(= 2\). Range \(=12 - 2=10\). So third option is true.

Step4: Analyze standard deviation

Since actual data has more deviation (e.g., \(0\) in actual for sum \(3\) while expected has \(2\)), standard deviation of actual is more. So fourth option is false.

Step5: Check medians and quartiles

Expected data: ordered \(1,1,2,2,3,3,4,4,5,5,5,6\) (for median and quartiles). Actual data: ordered \(0,1,1,3,3,3,4,5,5,6,6\) (different). So fifth option is false.

Answer:

The mean of the actual sums is more than the mean of the expected sums. The range of the sums is the same for both sets of data.