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. sum expected actual 2 1 1 3 2 0 4 3 5 5 4 5 6 5 6 7 6 5 8 5 3 9 4 3 10 3 4 11 2 3 12 1 1 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: Analyze symmetry

The expected data is symmetric (1 - 2 - 3 - 4 - 5 - 6 - 5 - 4 - 3 - 2 - 1), but the actual data is not symmetric. So "Both data sets are symmetrical" is false.

Step2: Calculate means

Expected mean: $\sum_{i = 2}^{12}i\times P(i)$, where $P(i)$ is the expected frequency divided by 36.
Actual mean: $\sum_{i = 2}^{12}i\times Q(i)$, where $Q(i)$ is the actual frequency divided by 36.
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 "The mean of the actual sums is more than the mean of the expected sums" is true.

Step3: Calculate ranges

Range = Maximum - Minimum. For both expected and actual, maximum = 12, minimum = 2, range = 10. So "The range of the sums is the same for both sets of data" is true.

Step4: Calculate standard - deviations

Standard deviation formula: $\sigma=\sqrt{\frac{\sum_{i}(x_{i}-\mu)^{2}f_{i}}{n}}$ where $x_{i}$ is the value, $\mu$ is the mean, $f_{i}$ is the frequency and $n$ is the total number of data points.
Calculating the standard - deviation for expected and actual data shows that the standard deviation of the actual sums is not less than that of the expected sums. So "The standard deviation of the actual sums is less than that of the expected sums" is false.

Step5: Calculate medians and quartiles

For expected data (36 data points), median is the average of 18th and 19th ordered values. For actual data (36 data points), median is the average of 18th and 19th ordered values.
Expected data: Arranging the frequencies in order, the median is 7.
Actual data: Arranging the actual frequencies in order, the median is not the same as the expected median. So "The medians and quartiles are the same in both sets" 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.