To determine if it is appropriate to use the methods for constructing a confidence interval for the proportion of days with unhealthy air quality, we need to check the conditions for using a normal approximation.
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We denote \(n\) as the sample size (200 days), and we'll define \(p\) as the true proportion of days with unhealthy air quality in the population. In the sample, there were 100 days with unhealthy air quality, thus the sample proportion \( \hat{p} = \frac{100}{200} = 0.5 \).
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The conditions that need to be satisfied for the normal approximation to be appropriate are:
- \(n \hat{p} \geq 10\)
- \(n (1 - \hat{p}) \geq 10\)
Now, let's check these conditions:
- \(n \hat{p} = 200 \times 0.5 = 100\) which is greater than 10.
- \(n (1 - \hat{p}) = 200 \times (1 - 0.5) = 200 \times 0.5 = 100\) which is also greater than 10.
Since both conditions are satisfied, we can conclude that it is appropriate to use the methods to construct a confidence interval for the proportion of days on which the air quality is unhealthy.
Therefore, the answer is:
A) Yes