Please show a step be step solution.

Problem 3 (12 points)

An unprepared student is about to take his first test of the semester. He is told that his test will be composed of 20 multiple choice questions, each with 4 possible answers, only one of which is correct.

a. What is the probability that the unprepared student randomly guesses at the correct response on ONE of the questions and gets the question correct?

b. What is the probability that the unprepared student randomly guesses at the correct response on ONE of the questions and gets the question wrong?

c. What is the probability that the student guesses at all 20 questions and gets at least one questions correct?

Problem 4 (9 points)

Spell-checking software catches non-word errors which result in a string of letters that is not a word, as when the is type teh. When undergraduates are asked to write a 250-word essay (without spell checking), the number X of non-word errors has the following distribution:

Errors (X)	0	1	2	3	4
Probability	0.3	0.25	0.2	0.17

a. In the table shown above the probability of exactly 4 errors is not given. Using what you know about probability distributions, what is the probability of exactly 4 errors?

1. b. What is the probability of at least 1 non-word error?

1. c. Find the expected value of this distribution. How many non-word errors can you expect from a typical paper written under the previously mentioned conditions?

Problem 5 (8 points)

Researchers conducted a study to determine whether there were significant differences in graduation rates between medical students admitted through special programs (such as affirmative action) and medical students admitted through the regular admissions criteria. It was found that the graduation rate was 84% for the medical students admitted through special programs.

a. If 15 of the students from the special programs are randomly selected, find the probability that exactly 12 of them graduated.

1. b.If15ofthestudentsfromthespecialprogramsarerandomlyselected,findtheprobabilitythatfewerthan10ofthemgraduated.

Problem 6 (16 points)

Assume that the lengths of pregnancies are normally distributed with a mean of 280 days and a standard deviation of 7 days.

a. What is the probability that a randomly selected mother will have a pregnancy lasting longer than 300 days?

1. b. What is the probability that a randomly selected mother will have a pregnancy lasting shorter than 268 days?

1. c. What is the probability that a randomly selected mother will have a pregnancy lasting between 272 and 284 days?

1. d. Find the 95^th percentile for the lengths of pregnancies.