A random sample of 1027 adults in a certain large country was asked "Do you pretty much think televisions are a necessity or a luxury you could do without?" Of the 1027 adults surveyed, 521 indicated that televisions are a luxury they could do without. Complete parts (a) through (e) below. (a) Obtain a point estimate for the population proportion of adults in the country who believe that televisions are a luxury they could do without. \[ \hat{\mathrm{p}}=0.507 \] (Round to three decimal places as needed.) (b) Verify that the requirements for constructing a confidence interval about $p$ are satisfied The sample __________ a simple random sample, the value of __________is ___which is _________10 , and the____________ ___________ less than or equal to $5 \%$ of the_____________ (Round to three decimal places as needed.)

Step 1 ：Calculate the point estimate for the population proportion (p̂) by dividing the number of successes (people who believe that televisions are a luxury they could do without) by the total number of trials (total number of people surveyed). So, \(\hat{p} = \frac{521}{1027} = 0.507\) (rounded to three decimal places)

Step 2 ：Check the requirements for constructing a confidence interval about p. The first requirement is that the sample is a simple random sample. This means that every individual in the population has an equal chance of being included in the sample. The problem states that it is a random sample, so this condition is met.

Step 3 ：The second requirement is that the value of np and n(1-p) are both greater than or equal to 10. This is to ensure that the sampling distribution of the sample proportion is approximately normal. Calculate np and n(1-p) as follows: \(np = 1027 * 0.507 = 520.869\) and \(n(1-\hat{p}) = 1027 * (1 - 0.507) = 506.131\). Both values are greater than 10, so this condition is also met.

Step 4 ：The third requirement is that the sample size is less than or equal to 5% of the population size. This is to ensure that the sample is not too large relative to the population, which could bias the results. Since the problem does not provide information about the population size, we cannot verify this condition. However, if we assume that the country is a large one (say, with a population of at least 20 million), then the sample size of 1027 would be less than 5% of the population size, and this condition would be met.

Step 5 ：Based on the information provided, we can say that the requirements for constructing a confidence interval about p are satisfied. \(\boxed{\text{The requirements for constructing a confidence interval about p are satisfied.}}\)