*Applied Statistics*

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**HW_4_1) Write the statement of the***Central Limit Theorem*(CLT). Interpret its meaning and give some arguments/examples of why is it important and how is it used in practice. What is so special about the sample average? Are there similar results that are valid for sample SD, Q_{1}, IQR, or any other statistic of the data?

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**HW_4_2) According to one US Government study approximately 22% of the American children under 6 live in households with income less then the poverty level. A random sample of 700 children under the age of 6 is selected for a study of learning in early childhood. Approximately how likely is that at least 250 of those children live in poverty?**

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**HW_4_3) Suppose a high speed network connection uses bits of 0's and 1's to transfer data between server's and client's computers. The network does on-the-fly correction of corrupted bits, but fails if there are more than 12 incorrect bits in 10,000. The chance that a single bit is incorrectly transmitter is 10-4, and the error rate for each bit is independent of the other bits. Data is significantly compressed and encrypted before transmission and decrypted and uncompressed upon arrival. This way 1 MB message is transmitted (approximately) as 100,000 bits. What is the compression ratio for the network? What is the chance that a 1 MB message is corrupted in the process of transferring the compressed data? [Think about Binomial model and identify the parameters of interest. Is Normal approximation valid in this situation?]**