50 kids and adults are surveyed about weather for hamburgers or hotdogs the results are shown in the table the table.
kids adults
hamburgers 8. 17
hot dogs. 12. 13
are being a kid and preferring hotdogs independent? explain how you know
To determine if being a kid and preferring hot dogs are independent, we need to calculate the probabilities and see if the probability of both events happening together is equal to the product of the probabilities of the two individual events.
Total number of kids = 8 (hamburger kids) + 12 (hot dog kids) = 20
Total number of adults = 17 (hamburger adults) +13 (hot dog adults) = 30
Total number of people = 20 (kids) + 30 (adults) = 50
Probability of being a kid (P(A)) = Total kids / Total people = 20 / 50 = 0.4
Probability of preferring hot dogs (P(B)) = Total hot dog preferences / Total people = 25 / 50 = 0.5
Probability of both being a kid and preferring hot dogs (P(A ⋂ B)) = Hotdog kids / Total people = 12 / 50 = 0.24
Now, if being a kid (A) and preferring hot dogs (B) are independent, then P(A ⋂ B) = P(A) * P(B).
Let's check if this holds true:
P(A) * P(B) = 0.4 * 0.5 = 0.2
Since P(A ⋂ B) ≠ P(A) * P(B) (0.24 ≠ 0.2), being a kid and preferring hot dogs are NOT independent events.
To determine if being a kid and preferring hotdogs are independent, we need to compare the observed frequencies with the expected frequencies.
First, let's calculate the total number of people surveyed:
Total = number of kids + number of adults = 50
Next, we can calculate the expected frequencies by multiplying the row and column totals and dividing them by the total number of people surveyed:
Expected frequency for kids who prefer hotdogs = (total number of kids * total number of people who prefer hotdogs) / Total
Expected frequency for adults who prefer hotdogs = (total number of adults * total number of people who prefer hotdogs) / Total
Expected frequency for kids who prefer hamburgers = (total number of kids * total number of people who prefer hamburgers) / Total
Expected frequency for adults who prefer hamburgers = (total number of adults * total number of people who prefer hamburgers) / Total
Next, we compare the observed frequencies with the expected frequencies. If the observed frequencies are close to the expected frequencies, then being a kid and preferring hotdogs are independent. If they are significantly different, then being a kid and preferring hotdogs are dependent.
To summarize, if the observed frequencies for kids who prefer hotdogs are close to the expected frequencies based on the overall proportions of kids and adults who prefer hotdogs, then being a kid and preferring hotdogs are independent.
To determine if being a kid and preferring hotdogs are independent, we need to compare the proportions of kids and adults who prefer hotdogs.
First, we calculate the proportions of kids and adults who prefer hotdogs:
- Proportion of kids who prefer hotdogs = Number of kids who prefer hotdogs / Total number of kids
- Proportion of adults who prefer hotdogs = Number of adults who prefer hotdogs / Total number of adults
Let's calculate these proportions using the given data:
- Proportion of kids who prefer hotdogs = 12 / 50 = 0.24 (or 24%)
- Proportion of adults who prefer hotdogs = 13 / 50 = 0.26 (or 26%)
Since the proportion of kids who prefer hotdogs (0.24) is not equal to the proportion of adults who prefer hotdogs (0.26), we can conclude that being a kid and preferring hotdogs are not independent.
If being a kid and preferring hotdogs were independent, the proportions of kids and adults who prefer hotdogs would be equal. In this case, the proportions are different, suggesting that there may be a relationship between being a kid and having a preference for hotdogs.