Question

In a local university, 10% of the students live in the dormitories. A random sample of 100 students is selected for a particular study. (a) What is the probability that the sample proportion (the proportion living in the dormitories) is between 0.169 and 0.175? (Round your answer to four decimal places.) (b) What is the probability that the sample proportion (the proportion living in the dormitories) is greater than 0.085 ? (Round your answer to four decimal places.)

Answer 1

To find the probabilities related to the sample proportion, we can use the normal approximation to the binomial distribution.

Step-by-step explanation:

To solve this problem, we can use the normal approximation to the binomial distribution.

(a) To find the probability that the sample proportion is between 0.169 and 0.175, we want to find P(0.169 ≤ p ≤ 0.175). We can convert this to a standard normal distribution by using the formula z = (p - P') / √((P' * (1 - P')) / n), where P' is the population proportion, p is the sample proportion, and n is the sample size. We can then use a z-table or a calculator to find the probability.

(b) To find the probability that the sample proportion is greater than 0.085, we want to find P(p > 0.085). Again, we can convert this to a standard normal distribution and use a z-table or a calculator to find the probability.

Answer 2

The probability of needing to contact four people to find a student living within five miles is 0.0641.

Step-by-step explanation:

Suppose that you are looking for a student at your college who lives within five miles of you. You know that 55 percent of the 25,000 students do live within five miles of you. You randomly contact students from the college until one says he or she lives within five miles of you. What is the probability that you need to contact four people?

To calculate the probability, we need to find the complement of the event that you find a student living within five miles of you on a specific contact.

The probability that a student doesn't live within five miles of you is 1 - 0.55 = 0.45. So, the probability that you need to contact four people is (0.45)^3 * 0.55 = 0.0641.