Question

assume that 30% of students at a university wear contact lenses. a random sample of 100 students is selected. a. describe the sampling distribution of the sample proportion of students who wear contacts. b. what is the probability that at most one third of this sample wear contacts?

Answer 1

The sampling distribution of the sample proportion of students who wear contacts is approximately normal. The probability that at most one third of the sample wear contacts can be calculated using the normal distribution. The probability that at most one third of this sample wear contacts is the cumulative probability associated with a z-score of -0.222.

Step-by-step explanation:

a. The sampling distribution of the sample proportion of students who wear contacts is approximately normal due to the large sample size of 100 students and the assumption of random sampling. The mean of this sampling distribution is equal to the population proportion, which is 30%. The standard deviation of the sampling distribution, also known as the standard error, can be calculated using the formula SE = sqrt((p*(1-p))/n), where p is the population proportion and n is the sample size. In this case, SE = sqrt((0.3*(1-0.3))/100) = 0.045.

b. To find the probability that at most one third of this sample wear contacts, we need to calculate the cumulative probability up to and including one third. This can be done using the normal distribution with a mean of 0.3 and a standard deviation of 0.045. Using a calculator or statistical software, we can find the cumulative probability associated with the z-score for one third, which is (1/3 - 0.3)/0.045 = -0.222. The probability that at most one third of this sample wear contacts is the cumulative probability associated with a z-score of -0.222.