Suppose that the proportion of students who used the Internet as their major resource for school in the past was 66%. A sample of 1000 students was taken and the number of students who used the Internet for their school during the past year was recorded. Let ��̂ be the proportion of students surveyed who used the Internet in the past year.

(A) What is the approximate distribution of ��̂ ?
(B) What is the probability that the sample proportion ��̂ exceeds 68%?
(C) What is the probability that the sample proportion lies between 64% and 68%?

To answer these questions, we can use the concept of sampling distribution and the Central Limit Theorem. The distribution of the sample proportion ȳ (read as "y-bar") is approximately normal when the sample size is large and random. The mean of the sample proportion is equal to the population proportion, and the standard deviation is given by the formula:

σ(ȳ) = sqrt((p * (1 - p)) / n)

where:
- σ(ȳ) is the standard deviation of the sample proportion,
- p is the population proportion (in this case, 0.66), and
- n is the sample size (in this case, 1000).

(A) The approximate distribution of ȳ is a normal distribution with mean p (the population proportion) and standard deviation σ(ȳ) as calculated above.

(B) To find the probability that the sample proportion ȳ exceeds 68% (0.68), we first need to standardize the value using the formula:

z = (x - μ) / σ(ȳ)

Where:
- z is the z-score,
- x is the value we want to find the probability for (0.68 in this case),
- μ is the mean of the distribution (which is p, the population proportion), and
- σ(ȳ) is the standard deviation of the sample proportion.

By calculating the z-score, we can then find the probability associated with that z-score using a standard normal distribution table or a statistical software.

(C) Similarly, to find the probability that the sample proportion ȳ lies between 64% (0.64) and 68% (0.68), we need to calculate the z-scores for both values and find the difference in probabilities.

It's worth noting that these calculations assume that the sample was taken randomly and that the conditions for the Central Limit Theorem are satisfied (large sample size and independence of observations).

Using these steps, you should be able to find the answers to (A), (B), and (C) using the given information.