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The table summarizes a sample of people asked if they live with cats and dogs. Are the two variables (living with a dog, living with a cat) likely independent? Why or why not?

| | Lives with a dog | Does not live with a dog |
|-------------------|------------------|--------------------------|
| Lives with a cat | 20 | 70 |
| Does not live with a cat | - | 1355 |

A. They are likely independent because dogs and cats are different animals.
B. They are likely not independent because the data shows most people live without a cat or a dog, and the fewest people live with both cats and dogs.
C. They are likely independent because there is an overlap in the marginal frequencies when considering a margin of error.
D. They are likely not independent because there is no overlap in the marginal frequencies, even when considering a margin of error.

Sagot :

GinnyAnswer
To determine if the two variables (living with a dog, living with a cat) are likely independent, we need to check the relationship between the observed frequencies and the expected frequencies under the assumption of independence.

Here are the steps we can follow:

1. Calculate the Marginal Totals:
- Total number of people who live with a cat: [tex]\(20 + 70 = 90\)[/tex]
- Total number of people who do not live with a cat: We do not have this information directly from the table.
- Total number of people who live with a dog: [tex]\(20\)[/tex] (lives with both a dog and a cat).
- Total number of people who do not live with a dog: [tex]\(70\)[/tex] (lives with a cat but not a dog).

2. Overall Total:
- Total number of people asked: [tex]\(20 + 70 + 1355 = 1445\)[/tex]

3. Calculate Expected Frequencies:
- If the two variables were independent, the proportion of people who live with a dog should be similar regardless of whether they live with a cat. We need to use the marginal totals to calculate expected frequencies for independence.

- Expected frequency for people living with both a cat and a dog:
[tex]\[ \text{Expected frequency} = \left( \frac{\text{Total who live with a dog}}{\text{Total surveyed}} \right) \times \left( \text{Total surveyed who live with a cat} \right) \][/tex]
- Without knowing the marginal totals for people who do not live with a cat and who do not live with a dog, precise calculations cannot be provided here. However, the general comparison is still possible.

4. Compare Observed and Expected Frequencies:
- Observed number of people living with both a cat and a dog is [tex]\(20\)[/tex].
- If the number of people living with both a cat and a dog deviates significantly from the expected frequency under independence, then the variables are not independent.

Given the data, it is clear that most people live without a cat or a dog (shown in the marginal frequency being much larger) and the fewest live with both (only 20 people). Thus, the observed frequencies suggest that people living with both a cat and a dog are relatively rare compared to other combinations.

From this, we can conclude:

- They are likely not independent because the data shows most people live without a cat or a dog and the fewest people live with both cats and dogs.
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