Digital communication networks or channels work by sending and receiving individual bits that can have values of either 0 or 1 . Due to noise during the transmission of a message (which is just a bunch of bits sent and received one after another), a single bit has probability
p
of being received erroneously (i.e. it was sent as a 0 , but received as a 1 , or sent as a 1 and received as a 0 ). A telecommunications company has deemed an error rate of
p>0.05
to be unacceptable and want to test the hypothesis that
p=0.05
against the alternative hypothesis that
p>0.05
. The test consists of sending a predetermined 32-bit-long message and counting the number of erroneous bits received at the other end. Define
X
as the number of erroneous bits received. In order to ensure an acceptable level of significance, the communication channel is deemed unacceptable if
X≥4
(note: this is an arbitrary choice). i. Clearly state the null and alternate hypotheses. Draw a diagram illustrating the critical region. ii. Determine the probability of committing a type I error. Describe, in words, what a type I error is for this scenario. [Note: your MATLAB code from earlier in the semester can be used to calculate probabilities from the Binomial distribution.] iii. Determine the probability of committing a type II error when testing against specific alternative hypotheses of
p=0.15
and
p=0.2
. Describe, in words, what a type II error is for this scenario. iv. In order to decrease the probability of both types of errors simultaneously, a longer message can be used for the test. When a 400-bit-long message is sent, the null hypothesis is rejected if 30 or more erroneous bits are received (note: this is an arbitrary choice to ensure an acceptable level of significance). Use the normal approximation to the Binomial distribution to approximate the probability of a type I error,
α
