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%?

(A) The approximate distribution of the sample proportion ��̂ can be described by the normal distribution. This assumption is based on the Central Limit Theorem, which states that for large sample sizes, the distribution of sample proportions will approximate a normal distribution regardless of the shape of the population distribution. In this case, since the sample size is 1000 (which is considered large), we can approximate the distribution of ��̂ to be normal.

(B) To calculate the probability that the sample proportion ��̂ exceeds 68%, we need to calculate the z-score for the given proportion and then find the corresponding probability using the standard normal distribution.

First, we need to find the standard deviation (sigma) of the sample proportion ��̂. The formula for the standard deviation of a sample proportion is:

sigma = sqrt((p * (1 - p)) / n)

where p is the proportion of students who used the Internet as their major resource (0.66), and n is the sample size (1000).

sigma = sqrt((0.66 * (1 - 0.66)) / 1000)

Now we can calculate the z-score:

z = (x - p) / sigma

In this case, x is the value we want to calculate the probability for (0.68), and p is the proportion of students who used the Internet as their major resource (0.66). Substituting in these values:

z = (0.68 - 0.66) / sigma

After calculating the value of sigma, substitute the value in the equation and calculate the z-score.

Using a standard normal distribution table or a calculator, you can find the probability corresponding to this z-score. This will give you the probability that the sample proportion ��̂ exceeds 68%.

(C) To calculate the probability that the sample proportion lies between 64% and 68%, we need to find two z-scores: one for 64% and one for 68%. Then we can find the probability between these two z-scores using the standard normal distribution.

Following the same steps as in part (B), calculate the z-scores for 64% and 68% using the given formula:

z_64 = (0.64 - 0.66) / sigma
z_68 = (0.68 - 0.66) / sigma

Once you have the z-scores, you can find the corresponding probabilities by finding the area under the standard normal distribution curve between these two z-scores. This will give you the probability that the sample proportion ��̂ lies between 64% and 68%.