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Answer :
Final answer:
The sample proportion p of adult Americans owning pets will approximately follow a normal distribution with expected value 0.36 and variance 64.512, due to the Central Limit Theorem.
Explanation:
Let's take n=280 as the sample size. We are looking at the distribution of the sample proportion p, which represents the proportion of adults in the sample who own a pet. Here are three key points:
- Expected Value: The expected value of p (E[p]) is equivalent to the true population proportion, in this case, 0.36 (36%).
- Variance: The variance of p (Var[p]) can be calculated using the formula npq, where n is the sample size, p is the probability of success (owning a pet) and q=1-p is the probability of failure. Using our values, this would be 280*0.36*0.64 = 64.512.
- Distribution: Since our sample size is large enough (>30), and the expected number of success and failures (np and nq) are both greater than 5, the distribution of p will approximately be a normal distribution (Central Limit Theorem).
To calculate the probability that more than 40% of sampled adults have a pet, we'd need to calculate the Z-score corresponding to p=0.40 and then find the area under the standard normal distribution curve greater than this Z-value. If this probability is less than 0.05 (5%), we consider this to be a statistically unusual occurrence, otherwise we deem it not to be unusual.
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