(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, Q1, IQR, or any other statistic of the data?
(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?
(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?]
\Ivo D. Dinov, Ph.D., Departments of Statistics and Neurology,
UCLA School of Medicine/