We will use the Sampling Distribution applet in StatCrunch to investigate properties of the sampling distribution of the proportion of first-generation college students at GMU from the previous problem. Remember, the given probability of first-generation students at Mason is 0.39.

Under Applets, open the Sampling distribution applet (box shown below). First, select Binary for the population. Next, to the right of "p:", enter 0.39, which is the probability of being a first-generation student at GMU. Then click on Compute! See image below.

a) Once the applet box is opened, enter 10 in the box to the right of the words "sample size" in the right middle of the applet box window (see image below). Then, at the top of the applet, click "1 time." Watch the resulting animation. When the sample is completed, copy and paste the entire applet box (using options copy) into your document.

b) Click Reset at the top of the applet. Then, click the "1000 times" to take 1000 samples of size 10. Copy and paste the applet image into your document.

c) Describe the shape of the Sample Proportions graph at the bottom of your image from part (b) in one sentence.

d) Use the Central Limit Theorem large sample size condition to determine if it is reasonable to define this sampling distribution as normal. Explicitly show these calculations in your answer.

e) Click Reset at the top of the applet. Type 100 in the sample size box. Then, click the "1000 times" to take 1000 samples of size 100. Copy and paste the applet image into your document.

f) Describe the shape of the Sample Proportions graph at the bottom of your image from part (e) in one sentence.

g) Why do you think that this graph from part (e) has the shape you described? Use the Central Limit Theorem large sample size condition to answer this question in one sentence. Explicitly show these calculations.

h) Using the image in part (e), write the values you obtained for the mean (in green) and the standard deviation (in blue). These values are found in the bottom right box labeled "Sample Prop. of 1s."

i) Compare the mean value (in green, found in part (h)) to the known population proportion in one sentence.

j) Now calculate the standard error of the sample proportion using p = 0.39 and n = 100 by hand. Type your "by hand" work and round your answer to four decimal places.

k) Compare the value in part (j) to the standard deviation (in blue) you obtained in part (h) in one sentence.

l) Use the sampling distribution defined by the Central Limit Theorem to calculate the probability that from a sample of 100 students at least 45% of students are first-generation students (use p = 0.39 and the standard error found in part (j)). First, draw a picture with the mean labeled, shade the area representing the desired probability, standardize, and use the Standard Normal Table (Table 2 in your text) to obtain this probability. Please take a picture of your hand drawn sketch and upload it to your document (if you do not have this technology, you may use any other method (i. e. Microsoft paint) to sketch the image). You must type the rest of your "by hand" work (e. g. the formula to calculate the z-value) to earn full credit.

m) Provide a StatCrunch Normal graph to verify the work in part (l) and interpret the resulting probability in context.