Packet I for Intro Stat Review Instructor: Katrina Anderson Name: Please show all work where applicable! You are free to use any source to fill out this document, however your work must be your own; it is to your benefit to work on these so that you understand them. If you work together make sure you and your groupmates are able to understand the results/answers. EMAIL ME if you have questions: kjanders@marymount.edu Version A 1. The mean of the sampling distribution of p̂ is: q (a) np; npq (d) p̂; p̂q̂ n (b) p̂; pq √ n (e) p; (c) p; pq n (f) p; and the variance is: pq √ n p pq n . (g) np̂; np̂q̂ (h) p̂; p̂q̂ n (i) none of the above. 2. For a discrete random variable, P (X ≥ x) = 1 − P (X ≤ x). True or False 3. is a statistic for central tendancy, which is thus a . (a) s; constant (d) µ; random variable (g) x̄; random variable (b) σ; random variable (e) s; random variable (h) µ; constant (c) x̄; constant (f) σ; constant (i) none of the above 4. By the Theorem, for any normal population, p̂ is: (a) normally distributed only if the sample size is greater than or equal to 5. (b) approximately normally distributed only if the sample size is greater than or equal to 30. (c) normally distributed only if the sample size is greater than or equal to 30. (d) approximately normally distributed no matter the sample size. (e) normally distributed no matter the sample size. (f) approximately normally distributed only if the sample size is greater than or equal to 5. (g) none of the above. 5. The probability density function describes a sample. True or False Page 2 Version A 6. NordicTrac manufactures several kinds of physical fitness machines, however they are most widely known for their treadmills. The life spans of treadmills are known to be normally distributed with a mean of 8 years and a standard deviation of 3.46 years. (a) What is the probability that the mean life span in a group of 10 randomly selected treadmills is between 7 and 11 years? (b) What is the probability that a randomly selected treadmill has a life span of between 12 and 14 years? (c) What is the probability that a randomly selected treadmill has a life span of more than 13 years? (d) The SLC is going to be under repair over the coming weeks and will be upgrading some of their equipment. They want to make sure that they get a warranty on the machines that will cover the equipment. They want to ensure that they will be able to have 95% of their new treadmills replaced under warranty. How many years should they try to get covered under warranty? Page 3 Version A 7. We wish to know how many hours a typical Baylor student spends at the SLC during a given day. Suppose the probability distribution is as follows: X 0 0.5 1 1.25 1.5 2 4 F(x) 0.37 0.67 0.75 a 0.92 0.99 b (a) If the P (X = 1.25) = 0.12 what is the value of a? (b) Find the PMF. (c) What is the random variable? (d) Find and interpret E(X). (e) Find the probability that a randomly selected student works out at the SLC for at least 1.25 hours in a given day. Page 4 Version A 8. An athlete is trying to randomize his workouts between cardio and weight training. He decides that at the start of each workout he will flip a coin and either do cardio (heads) or weight lifting (tails) for that session. After 20 workouts, he had done 14 cardio sessions. The following JMP output could be helpful. (a) What is the random variable? (b) What is the distribution of X? (c) How many days of cardio would we expect him to complete after 20 workout sessions? (d) What is the probability that he does between 11 and 15 cardio workout sessions (inclusive)? (d) What is the probability that he does more than 13 cardio workout sessions? Page 5 Version A 9. Fill in the blank with the appropriate symbol or term. (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) x̄ : : sample variance σ: : population size x̃ : : population mean n: : population proportion s: : population median p̂ : : population variance 10. Suppose there are two tests which measure a person’s level of physical fitness; both tests are measured in such a way that a higher score means better physical fitness. Which test score relates to a more physically fit person, a physical fitness score of 92 on a test with a mean of 71 and a standard deviation of 15 or a score of 688 on a test with a mean of 493 and a standard deviation of 150? Page 6 Version A 11. Find the following probabilities/values: (a) Z0.28 (b) P (Z ≥ 0.1245) (c) P (−2.1511 < Z < 2.1872) 12. Use the following graphic to answer the questions below. i. What type of distribution is shown? (a) right skewed (b) symmetric (c) bell shaped (d) left skewed (e) bimodal (f) uniform (g) normal (h) binomial ii. Which measure of central tendency should we use? (a) Standard Deviation (b) Range (c) Median (d) Mode (e) Q1 (f) Mean Page 7 (g) Variance (h) IQR (i) Q3 Version A 13. Suppose it is known that the time it takes to run a half marathon has a uniform distribution from 65 to 195 minutes. (a) What is the probability distribution function? (b) What is the probability that the time to run a half marathon is more than 2 hours? 14. The distribution for the BMI of Baylor students is given below. Which of the following answer choices are possible percentiles for the data value of X = 28? Circle all answers that apply. (a) 15% (d) 10% (g) 65% (b) 85% (e) 45% (h) 35% (c) 90% (f) 55% (i) 99% Page 8 Version A Formulae Q3 + 1.5 · IQR E[X] = N X xi f (xi ) P (X = x) = n! px (1 − p)n−x (n − x)!x! Q1 − 1.5 · IQR i=1 " V [X] = N X i=1 Page 9 # x2i f (xi ) − µ2 Packet 1 Review X -8 0 3 4 5 6 f(x) 0.12 0.34 A 0.41 0.02 0.03 a) Find the value of A b) Find the CDF c) What is the probability that the random variable X falls between 2 and 5 (inclusive)? i. Using the PDF: ii. Using the CDF: X -8 0 3 4 5 6 f(x) 0.12 0.34 A 0.41 0.02 0.03 a) Find the value of A 1- (0.03 + 0.02 + 0.41 + 0.34 + 0.12) = 1 - 0.92 = 0.08 b) Find the CDF X -8 0 3 4 5 6 F(x) 0.12 0.46 0.54 0.95 0.97 1.0 c) What is the probability that the random variable X falls between 2 and 5 (inclusive)? i. Using the PDF: f(2) + f(3)+f(4)+f(5) = 0.08 + 0.41 + 0.02 = 0.51 ii. Using the CDF: F(5) – F(0) = 0.97 – 0.46 = 0.51 A man claims to have extrasensory perception (ESP). As a test, a fair coin is flipped 12 times, and the man is asked to predict the outcome in advance. He gets 8 out of 12 correct. a) What is the random variable? b) What is the distribution of the random variable? c) What is the probability that he predicts between 8 and 10 flips correctly? d) How many correct predictions would we expect him to make if he were simply randomly guessing the outcome for all 12 flips? A man claims to have extrasensory perception (ESP). As a test, a fair coin is flipped 12 times, and the man is asked to predict the outcome in advance. He gets 8 out of 12 correct. a) What is the random variable? Guesses of the coin flip result b) What is the distribution of the random variable? Binomial c) What is the probability that he predicts between 8 and 10 flips correctly? 𝑃 8 < 𝑥 < 10 = 𝑃 𝑥 = 9 = 𝐵𝐼𝑁𝑂𝑀. 𝐷𝐼𝑆𝑇 9,12,0.5, 𝐹𝐴𝐿𝑆𝐸 = 0.0537 OR (partial credit) 𝑃 8 ≤ 𝑥 ≤ 10 = 𝑃 𝑥 225 = 𝑃 𝑧 > 𝑥−𝜇 𝜎 =𝑃 𝑧> 225−219.2 23.2 = 𝑃 𝑧 > 0.25 = 1 − P z ≤ 0.25 = 1 − 𝑁𝑂𝑅𝑀. 𝐷𝐼𝑆𝑇 0.25, 0, 1, 𝑇𝑅𝑈𝐸 = 0.4013 OR = 1 − 𝑁𝑂𝑅𝑀. 𝐷𝐼𝑆𝑇 225, 219.2, 23.2, 𝑇𝑅𝑈𝐸 = 0.4013 b) If a random sample of 20 individuals is selected at random, find the probability that their mean number of pounds of meat consumed the previous year is more than 223 pounds. ҧ 𝑥−𝜇 𝑛 𝑃 𝑥ҧ > 223 = 𝑃 𝑧 > 𝜎/ =𝑃 𝑧> 223−219.2 23.2/ 20 = 𝑃 𝑧 > 0.7325 = 1 − P z ≤ 0.7325 = 1 − 𝑁𝑂𝑅𝑀. 𝐷𝐼𝑆𝑇 0.7325, 0, 1, 𝑇𝑅𝑈𝐸 = 0.2319 OR = 1 − 𝑁𝑂𝑅𝑀. 𝐷𝐼𝑆𝑇 223, 219.2, 23.2 𝑠𝑞𝑟𝑡 20 , 𝑇𝑅𝑈𝐸 = 0.2319 Your favorite band is in town and you plan on attending their live show this weekend. Suppose the length of time the band plays during a show has a uniform distribution from 50 to 75 minutes (based on previous shows). a) What does the random variable, X, represent? b) What is the distribution for X? c) What is the probability that the band will play for over 65 minutes? d) If the band starts playing at 10:00 PM, but you have to leave by 11:00 PM, what is the probability you'll catch the entire show (assuming the band doesn't take any breaks)? e) What is the probability that the band will play for over an hour and a half? Your favorite band is in town and you plan on attending their live show this weekend. Suppose the length of time the band plays during a show has a uniform distribution from 50 to 75 minutes (based on previous shows). a) What does the random variable, X, represent? X: length of time the band plays b) What is the distribution for X? Uniform distribution c) What is the probability that the band will play for over 65 minutes? 1 1 1 𝑓 𝑥 = = = This means that the probability of playing for any distinct time 𝑚𝑎𝑥−𝑚𝑖𝑛 75−50 25 from 50 to 75 minutes has an equal probability of happening. You can also think of it that there are 1 out of 25 options to play for 50 minutes, or 51, or 52… So if we want to know the probability of playing over a range, we can simply find out the number of distinct minutes that are in that range (75-65)=10 and multiply! 𝑙𝑒𝑛𝑔𝑡ℎ 10 10 𝑃 𝑥 > 65 = = = = 0.4 𝑜𝑟 40% 𝑚𝑎𝑥 − 𝑚𝑖𝑛 75 − 50 25 d) If the band starts playing at 10:00 PM, but you have to leave by 11:00 PM, what is the probability you'll catch the entire show (assuming the band doesn't take any breaks)? You will make the entire show IF the band plays for at most 60 minutes, so: 𝑙𝑒𝑛𝑔𝑡ℎ 60−50 10 𝑃 𝑥 < 60 = = = = 0.4 𝑜𝑟 40% e) 𝑚𝑎𝑥−𝑚𝑖𝑛 75−50 25 What is the probability that the band will play for over an hour and a half? By the rules of probability, this is outside the range of possible x values, the probability is 0.
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