Let's go through each problem step-by-step:
4. Find the mode:
To find the mode, we look for the score with the highest frequency. The frequency table is as follows:
- Score: 0, Frequency: 2
- Score: 1, Frequency: 3
- Score: 2, Frequency: 4
- Score: 3, Frequency: 2
- Score: 4, Frequency: 7
- Score: 5, Frequency: 2
The highest frequency is 7, which corresponds to the score 4. Therefore, the mode is:
Answer: C. 4
5. Find the mean score:
To find the mean score, we calculate the weighted average of the scores using their frequencies.
[tex]\[ \text{Mean} = \frac{(\sum \text{Scores} \times \text{Frequencies})}{\sum \text{Frequencies}} \][/tex]
The total sum of the scores weighted by their frequencies is:
[tex]\[ 0 \times 2 + 1 \times 3 + 2 \times 4 + 3 \times 2 + 4 \times 7 + 5 \times 2 = 0 + 3 + 8 + 6 + 28 + 10 = 55 \][/tex]
The total number of frequencies is:
[tex]\[ 2 + 3 + 4 + 2 + 7 + 2 = 20 \][/tex]
Therefore, the mean score is:
[tex]\[ \text{Mean} = \frac{55}{20} = 2.75 \][/tex]
Answer: d. 2.75
6. The median score is:
To find the median, we need to list all scores in ascending order and then find the middle value. The scores listed with their frequencies are:
[tex]\[ 0, 0, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5 \][/tex]
The median is the value in the middle of this list. Since there are 20 scores, the middle values are the 10th and 11th scores:
[tex]\[ 3 \text{ and } 3 \][/tex]
The median is:
[tex]\[ \text{Median} = 3 \][/tex]
Answer: c. 3
7. Convert 89 in base 10 to a number in base 2:
89 in binary is:
[tex]\[ 1011001 \][/tex]
Answer: d. 1011001
8. Evaluate [tex]\( \left(20_{\text{three}}\right)^2 - \left(11_{\text{three}}\right)^2 \)[/tex] in base three:
We need to evaluate the expression in base 3:
[tex]\[ 20_{\text{three}} = 2 \times 3^1 + 0 \times 3^0 = 6_{\text{ten}} \][/tex]
[tex]\[ 11_{\text{three}} = 1 \times 3^1 + 1 \times 3^0 = 4_{\text{ten}} \][/tex]
[tex]\[ \left(6_{\text{ten}}\right)^2 - \left(4_{\text{ten}}\right)^2 = 36 - 16 = 20_{\text{ten}} \][/tex]
Now, converting 20 from base 10 to base 3:
[tex]\[ 20_{\text{ten}} = 202_{\text{three}} \][/tex]
Answer: c. 202
9. If [tex]\( Y = 23_{\text{five}} + 101_{\text{three}} \)[/tex], find [tex]\( Y \)[/tex], leaving your answer in base 2:
First, convert the numbers from their respective bases to base 10:
[tex]\[ 23_{\text{five}} = 2 \times 5 + 3 = 10 + 3 = 13_{\text{ten}} \][/tex]
[tex]\[ 101_{\text{three}} = 1 \times 3^2 + 0 \times 3^1 + 1 \times 3^0 = 9 + 0 + 1 = 10_{\text{ten}} \][/tex]
Add these two numbers in base 10:
[tex]\[ Y = 13 + 10 = 23_{\text{ten}} \][/tex]
Convert 23 from base 10 to base 2:
[tex]\[ 23_{\text{ten}} = 10111_{\text{two}} \][/tex]
Answer: b. 10111
10. Find the mean deviation of [tex]\( 6, 7, 8, 9, 10 \)[/tex]:
To find the mean deviation, we first find the mean of the data set:
[tex]\[ \text{Mean} = \frac{6 + 7 + 8 + 9 + 10}{5} = \frac{40}{5} = 8 \][/tex]
Next, find the absolute deviations from the mean for each data point:
[tex]\[ |6 - 8| = 2 \][/tex]
[tex]\[ |7 - 8| = 1 \][/tex]
[tex]\[ |8 - 8| = 0 \][/tex]
[tex]\[ |9 - 8| = 1 \][/tex]
[tex]\[ |10 - 8| = 2 \][/tex]
The mean deviation is the average of these absolute deviations:
[tex]\[ \text{Mean Deviation} = \frac{2 + 1 + 0 + 1 + 2}{5} = \frac{6}{5} = 1.2 \][/tex]
Answer: a. 1.2
