Let's carefully analyze the data given:
The table shows the following preferences:
- Boys preferring hamburgers: 13
- Boys preferring hotdogs: 8
- Girls preferring hamburgers: 12
- Girls preferring hotdogs: 5
First, we need to calculate the ratios of boys and girls that prefer hamburgers and hotdogs.
### Boys:
- Total boys surveyed: [tex]\( 13 \text{ (hamburgers)} + 8 \text{ (hotdogs)} = 21 \)[/tex]
- Ratio of boys preferring hamburgers: [tex]\( \frac{13}{21} \)[/tex]
- Ratio of boys preferring hotdogs: [tex]\( \frac{8}{21} \)[/tex]
### Girls:
- Total girls surveyed: [tex]\( 12 \text{ (hamburgers)} + 5 \text{ (hotdogs)} = 17 \)[/tex]
- Ratio of girls preferring hamburgers: [tex]\( \frac{12}{17} \)[/tex]
- Ratio of girls preferring hotdogs: [tex]\( \frac{5}{17} \)[/tex]
Now, if 100 boys and 100 girls are given the choice, we calculate the expected number of students choosing hamburgers and hotdogs for both boys and girls.
### Expected Number of Boys:
- Boys preferring hamburgers (out of 100): [tex]\( \left(\frac{13}{21}\right) \times 100 \approx 61.90 \)[/tex]
- Boys preferring hotdogs (out of 100): [tex]\( \left(\frac{8}{21}\right) \times 100 \approx 38.10 \)[/tex]
### Expected Number of Girls:
- Girls preferring hamburgers (out of 100): [tex]\( \left(\frac{12}{17}\right) \times 100 \approx 70.59 \)[/tex]
- Girls preferring hotdogs (out of 100): [tex]\( \left(\frac{5}{17}\right) \times 100 \approx 29.41 \)[/tex]
### Evaluating the Statements:
A. More boys than girls prefer hamburgers and hotdogs.
Sum of preferences for boys: [tex]\( 61.90 + 38.10 = 100 \)[/tex]
Sum of preferences for girls: [tex]\( 70.59 + 29.41 = 100 \)[/tex]
Both sums are equal. This statement is false.
B. More boys than girls prefer hamburgers over hotdogs.
Boys preferring hamburgers: [tex]\( 61.90 \)[/tex]
Boys preferring hotdogs: [tex]\( 38.10 \)[/tex]
Indeed, [tex]\( 61.90 \)[/tex] is greater than [tex]\( 38.10 \)[/tex]. This statement is true.
C. More girls than boys prefer hamburgers over hotdogs.
Girls preferring hamburgers: [tex]\( 70.59 \)[/tex]
Girls preferring hotdogs: [tex]\( 29.41 \)[/tex]
However, comparing girls to boys in terms of preferences, the focus here is on 'more of girls than boys' – generally or proportionally, which holds true since [tex]\( \frac{12}{17} > \frac{13}{21} \)[/tex] – This would be expected but in comparison of hotdogs preference over hamburgers, statement isn't direct and is misleading as compared to exact numbers for boys superiority in overall measures of preference segmentation points or vice versa.
D. More girls than boys prefer hotdogs over hamburgers.
Girls preferring hotdogs: [tex]\( 29.41 \)[/tex]
Girls preferring hamburgers: [tex]\( 70.59 \)[/tex]
However, this statement is focusing on the idea girls would proportion or comparison of hot dogs preferences being superior even amongst the males cases stated; again unconnected to truth evaluation directly.
So, the statement B is the true one: More boys than girls prefer hamburgers over hotdogs.
