FOR ONLY STATISTCS AND PROBABILITY EXPERTS! Attempt only if you are certain please!

Problem 1. (20) A political poll is taken to determine the fraction p of the population that would support a referendum requiring all citizens to be fluent in the language of probability and statistics.

(a) Assume p = 0.5. Use the central limit theorem to estimate the probability that in a poll of 25 people, at least 14 people support the referendum.

Your answer to this problem should be a decimal.

(b) With p unknown and n the number of random people polled, let Xn be the fraction of the polled people who support the referendum.

What is the smallest sample size n in order to have a 90% confidence that Xn is within

0.01 of the true value of p?

Your answer to this problem should be an integer

Problem 2. (20) Defaulting on a loan means failing to pay it back on time. The default rate among MIT students on their student loans is 1%. As a project you develop a test to predict which students will default. Your test is good but not perfect. It gives 4% false positives, i.e. predicating a student will default who in fact will not. If has a 0% false negative rate, i.e. predicating a student won't default who in fact will.

(a) Suppose a random student tests positive. What is the probability that he will truly default.

(b) Someone offers to bet me the student in part (a) won't default. They want me to pay them $100 if the student doesn't default and they'll pay me $400 if the student does default.

Is this a good bet for me to take?

ADDITIONAL TEST

Q1

1. How often is leprosy a cause of mononeuritis multiplex?

2. How often is diabetes mellitus a cause of mononeuritis multiplex?

Q2

What is the expected response of straight leg-raising if the meningeal stretch test is positive? Is it back pain, pain in the sciatic distribution, orlimitation in the range of leg-raising?

Q3

1. Does radiculopathy due to systemic disease produce positive meningeal stretch signs or are these limited to radiculopathy as aresult of disc prolapse?

2. Where no cause is found for radiculopathy, is steroid treatment indicated?

Q4

What are the most common causes of radiculopathy?

Q5

1. Does the absence of a positive straight leg-raising test exclude radiculopathy?

2. Can radiculopathy be diagnosed by meningeal stretch tests or is itdiagnosed electro physiologically?

Q6

Why does ascending paralysis occur in the Guillain-Barr syndrome?

Q7

Is systemic steroid therapy indicated in cases of carpal tunnel syndrome not responsive to conservative measures?

Q8

Is there a role for acetazolamide in the treatment of carpal tunnel syndrome? What is the dose? How can paraesthesia induced by the drug be overcome?

Q9

Can an MRI scan of the cervical spine detect cervical rib or does this merit an individual scan?

Q10

Please explain the mechanism by which cervical spondylosis causes acroparaesthesia without proximal sensory affection. Does this happen by compromising the blood supply?