To check whether the sample size is big enough to expect at least 10 successes and at least 10 failures, we can use the sample size \( n \) and the proportion of successes \( p \).
In this case, we have:
- \( n = 100 \) (total number of adults in the sample)
- \( x = 7 \) (number of adults with allergies)
Now, let's determine \( p \) (the proportion of successes, i.e., adults with allergies):
\[ p = \frac{x}{n} = \frac{7}{100} = 0.07 \]
Next, we calculate the expected number of successes (\( n \cdot p \)) and failures (\( n \cdot (1 - p) \)):
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Expected number of successes: \[ n \cdot p = 100 \cdot 0.07 = 7 \]
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Expected number of failures: \[ n \cdot (1 - p) = 100 \cdot (1 - 0.07) = 100 \cdot 0.93 = 93 \]
Now we can check the success-failure condition:
- We expected 7 successes (adults with allergies), which is less than 10.
- We expected 93 failures (adults without allergies), which is greater than 10.
Since we have fewer than 10 expected successes, the success-failure condition is not met.
Thus, the conclusion is that the sample size is not big enough to meet the success-failure condition.