Question

Let's say you want to poll a random sample of 150 students on campus to see if they prefer to take online classes. Of course, if you took an actual poll you would only get one number (your sample proportion, p-hat). But, imagine all the possible samples of 150 students that you could draw and the imagined histogram of all the sample proportions from those samples. 1. What shape would the histogram of all the possible sample proportions (p-hat's) have? 2. Where would the center of that histogram be? (This answer should be a description in words based on what we know about p-hat sampling distributions.) Now, given the information that about 35% of students actually prefer to take classes online respond to the following: 3. Discuss the conditions necessary to use the normal model here and explain whether or not they are met. 4. If you were to use the normal model for this p-hat sampling distribution, what would be the parameters (mean and standard deviation) for your model

Answer 1

Histogram is created using X on P(x), Binomial distribution is a bell shaped because the sample size is large enough for the sample data points to normally distributed.  

The center of the histogram is the average of the X values, that is equal to 75.

Mean and standard deviation of the binomial distribution models are 52.5 and 5.8417 (refer cell:B8 and B10). You can check for the average of X and standard deviation of X will be same, because the data is normally distributed.

Conditions to use normal model for this model is met and they are listed below:

The number of samples should be more than or equal to 30 numbers. There are 44 sample points from 0.0001 to 0.0001 probability density distribution. That means starting from X=32 to X=75, there are 44 sample data points.

Each student's response is independent of the any other student on campus.

Every student on campus either interested to do online course or interested to do a class room study. No student can do both at a time for the same subject. there are two possible outcomes for each student, but the event of interest shall be either one of them

The true population proportion for each student polled and it is constant for the entire population of the target students in the campus. P-hat-proportion of student interested to do online course is 0.35 and not interest to do online course is 0.65 is constant between the students and among all the students in the campus.