Lesson 5: Probability Distributions NAME Due on Monday midnight Answer the following questions showing all work. If you use Minitab Express include the appropriate output (copy + paste) along with an explanation. You need to download the Standard Normal Table to answer the question 4. Output without explanation will not receive full credit. Round all answers to 3 decimal places. If you have any questions, post them to the course discussion board. 1. [30 points] In Spring 2016 a sample of 523 Penn State World Campus STAT200 students were asked how many languages they can speak. Below is the probability distribution of their responses. X = Languages Probability 1 .7973 2 .1663 3 4 .0077 5 .0019 A. Is number of languages spoken a discrete or continuous variable? Explain why. B. What proportion of students in this sample can speak exactly 3 languages? C. Compute the mean of X [i.e., E(X)]. D. Compute the standard deviation of X. E. Complete the cumulative probability row of the following table: X = Languages 1 2 3 4 5 Probability .7973 .1663 ??? .0077 .0019 Cumulative Probability F. If you were to randomly select one student from this sample, what is the probability that he or she would speak more than one language? G. Suppose we are considering only the students who can speak more than one language. Then, for those students, what is the probability that they can exactly speak four languages [Hint: This is a conditional probability. You may want to first write it in the form of P (A|B), where A and B are events concerning X, and then use the definition of conditional distribution from the previous lesson to solve. You can use your answers in (g) to answer this question. ] Lesson 5: Probability Distributions NAME Due on Monday midnight 2. [20 points] Binomial Random Variables Definitions In each part, indicate (1) what is the variable of interest (X) AND (2) whether this variable is discrete or continuous AND (3) whether it is binomial or not AND (4) if it is binomial, give values for n and p. [Hint: think of the number of outcomes are countable (discrete) and the criteria for Binomial random variable] A. Number of times a \"head\" is flipped in 10 flips of a coin B. Time to complete a 60 question multiple choice test C. Number of correct answers on a 30 question multiple choice test for somebody who randomly guesses at every question. There are four answer choices for each question, D. A woman buys a lottery ticket every week for which the probability of winning anything at all is 1/10. She continues to buy them until she has won 3 times. X = the number of tickets she buys. 3. [30 points] Binomial Random Variables Probability Distributions Use Minitab Express to answer the following set of questions. Be sure to include all relevant output and clearly identify your answers by writing a sentence. A class is taking a test that consists of 40 true/false questions. The instructor made a mistake and printed the quiz for a much more advanced class. The students don't know any of the content and they randomly guess at every question without event reading them. A. Explain why this is a binomial random. B. What are the values of the parameters n and p? C. What is the mean quiz score? D. What is the standard deviation of the quiz scores? E. In order to pass, a student needs to get 24 or more questions correct. What is the probability that a student will get 24 or more questions correct? Calculate it by hand. F. In order to get an A, a student needs to get 36 or more questions correct. What is the probability that a student will get 36 or more questions correct? Calculate it by hand. G. What is the probability that a student will get all 40 questions correct? Calculate it by hand. Lesson 5: Probability Distributions NAME Due on Monday midnight H. Confirm your answer in part E using software, and paste your output here. 4. [20 points] Continuous Random Variables (Normal Distribution) For those intending to major in Business, quantitative GRE scores are normally distributed with mean 120 and variance 36. Let Y denote a randomly selected quantitative GRE score. Use this to answer the following questions (Note: in order to answer following questions, you need to know the standard deviation using the formula: s= s 2 ) A. [2 pts] Use software to find P(Y 105). Be sure to paste your output. Minitab: Calc > Probability Distributions > Normal. Minitab Express: Statistics > Probability Distributions > CDF/PDF > Cumulative (CDF). B. [2 pts] Write a sentence interpreting the value you found above. You should write the sentence so that it makes sense to someone not taking a statistics course. C. [2 pts] A certain university will only consider admitting applicants who have a quantitative GRE of at least 105. Use your answer from part a to find the probability that a randomly selected Business applicant will be considered for admittance. D. [2 pts] Use software to find the probability that an intended Business major scores between 110 and 130 on the quantitative GRE. Paste the output. [ Hint: P(110 < Y 130) = P(Y 130) - P(Y 110)] E. [8 pts] Using the Normal Probability table to answer the following questions. i. ii. iii. F. iv. Calculate the z-score of Y = 105. Round to two decimal places. [Hint: See the \"Finding Cumulative Probabilities\" page of the online notes.] Use the normal probability table and find your answer to part i to find P(Y 105) if we know a student's standardized score (z score) is 1, what is quantitative GRE scores for this student? [Hint: you need to transform z score to the raw score] Find the probability or proportion of scores between 100 and 126. [2 pts] A prestigious university will only considering admitting Business students who scored in at least the 80th percentile of the quantitative GREs. What is the minimum score would you need to get in order to be considered for admittance into the university? You may use software or the normal probability table to find the answer. [Hint: This is an inverse probability. You want to find a value y such that P(Y y) = .80] Lesson 5: Probability Distributions NAME Due on Monday midnight G. [2 pts] The program that you want to enroll in requires a total score in the top 3% of the population. What is the minimum score that you need to obtain to be admitted