Q$28.$ Spam filters try to sort your e$-$mails, deciding which are real messages and which are unwanted. One method used is a point system. The filter reads each incoming e$-$mail and assigns points to the sender$,$ the subject, key words in the message, and so on$.$ The higher the point total, the more likely it is that the message is unwanted. The filter has a cutoff value for the point total; any message rated lower than that cutoff passes through to your inbox, and the rest, suspected to be spam, are diverted to the junk mailbox. We can think of the filter's decision as a hypotesis test. The null hypothesis is that the e$-$mail is a real message and should go to your inbox. A higher point total provides evidence that the message may be spam; when there's sufficient evidence, the filter rejects the null. classifying the message as junk. This usually works pretty well, but, of course, sometimes the filter makes a mistake.

Q$30.$ Consider again the points$-$based spam filter described in exercise $28.$ When the points assigned to various components of an e$-$mail exceed the cutoff value you've set, the filter rejects its null hypothesis $($that the message is real$)$ and diverts that e$-$mail to a junk mailbox.

a$)$ In this context, what is meant by the power of the test?

b$)$ What could you do to increase the filter's power?

c$)$ What is the disadvantage of doing that?