One of the many wonderful things about studying statistics is that graduate programs in statistics often pay their graduate students, which means that many graduate students in statistics are able to attend graduate school tuition free with an assistantship or fellowship. In 2009, there were 82 US statistics or biostatistics doctoral programs for which enrollment data were available. The dataset StatisticsPhD lists all these schools together with the total enrollment of full-time graduate students in each program in 2009.

(a) Use StatKey or other technology to select a random sample of 10 of the 82 enrollment values. Indicate which values you've selected and compute the sample mean.

(b) Repeat part (a) by taking a second sample and calculating the mean.

(c) Find the mean enrollment for the entire population of these 82 graduate programs. Use correct notation for your answer. Comment on the accuracy of using the sample means found in parts (a) and (b) to estimate the population mean.

• (d) Give a rough sketch of the sampling distribution if we calculate many sample means taking samples of size \(n=10\) from this population of enrollment values. What shape will it have and where will it be centered?