Nichols-Assignment-D

pdf

School

University Of Arizona *

*We aren’t endorsed by this school

Course

376

Subject

Mechanical Engineering

Date

Jan 9, 2024

Type

pdf

Pages

Uploaded by ChancellorRockJaguar38

McKenna Nichols BME 376: Assignment D – Probability Distributions • This assignment is focused on probability distributions. All answers should be entered in the space provided below each question. • Type equations (using in-built Word features) and text. Avoid screenshots or images. • Upload a completed word document to D2L. 1. [4 points] Approximately 12% of the US population smokes cigarettes. A local government (county) believed their community had a lower smoker rate and commissioned a survey of 400 randomly selected individuals. What is the probability of observing more than 5 smokers in the sample? [2 points] Approach A (compute manually i.e., without R) 𝑷(𝒙) = ( 𝒏 𝒙 ) ? 𝒙 ? 𝒏−𝒙 𝑃(1) = ( 400 1 ) 0.12 1 0.88 400−1 = 3.3871 ∗ 10 −21 𝑃(2) = ( 400 2 ) 0.12 2 0.88 400−2 = 2.30938 ∗ 10 −22 𝑃(3) = ( 400 3 ) 0.12 3 0.88 400−3 = 2.09944 ∗ 10 −23 𝑃(4) = ( 400 4 ) 0.12 4 0.88 400−4 = 2.14715 ∗ 10 −24 𝑃(5) = ( 400 5 ) 0.12 5 0.88 400−5 = 2.34235 ∗ 10 −25 1 − (𝑃(1) + 𝑃(2) + 𝑃(3) + 𝑃(4) + 𝑃(5)) = 1 − (3.64141 ∗ 10 −21 ) = 1 [1 point] Approach B (use R) # include R commands here > 1 - pbinom(5, size = 400, p = 0.12) # paste R outputs here 1 What is the probability of observing less that 60 smokers in the sample? [1 point] Use R (manual approach not needed) # include R commands here > pbinom(59, size = 400, prob = 0.12) # paste R outputs here 0.9584233

2. [6 points] Based on census data, it is estimated that the percentage of individuals with no high school education/diploma in Southern Arizona is 23%. Suppose we select three individuals at random from the Southern Arizona community. What is the probability that none of the three individuals have high school diploma? [1 point] Approach A (compute manually i.e., without R) ( 3 3 ) = 3! 3! ∗ 3! = 1 6 𝑃(3) = ( 3 3 ) 0.23 3 0.77 3−3 = 0.012167 [1 point] Approach B (use R) # include R commands here > dbinom(3, size = 3, p = 0.23) # paste R outputs here 0.012167 What is the probability that at least two individuals have no high school diploma? [1 point] Approach A (compute manually i.e., without R) ( 3 1 ) = 3! 1! ∗ 3! = 1 𝑃(3) = ( 3 1 ) 0.23 1 0.77 3−1 = 0.0136367 [1 point] Approach B (use R) # include R commands here > 1 - pbinom(1, size = 3, p = 0.23) # paste R outputs here 0.134366 What is the probability that all three individuals have high school diploma? [1 point] Approach A (compute manually i.e., without R) ( 3 0 ) = 3! 0! ∗ 3! = 1 𝑃(3) = ( 3 0 ) 0.23 0 0.77 3−0 = 0.456533 [1 point] Approach B (use R) # include R commands here > dbinom(0, size = 3, p = 0.23) # paste R outputs here 0.456533

3. [9 points] The amount of weight gained during a pregnancy is an important indicator of the health of the mother as well as the baby. Learn more about the recommended weight gain during pregnancy and its implications to maternal and infant health here . Based on data from a birth data registry in Yuma County, the amount of weight gained during pregnancy (in pounds) was approximately normally distributed with a mean of 30.2 pounds and a standard deviation of 13.8 pounds. What is the probability that a randomly selected mother in Yuma County gained less than 20 pounds (i.e., less than recommended amount of weight gain)? [2 points] Approach A (compute manually i.e., without R) 𝑍 = 𝑥 − ? 𝜎 𝑍 = 20 − 30.2 13.8 = −0.73 Using the negative Z-table, the probability of this with the above z-score is 0.2327 [1 point] Approach B (use R) # include R commands here > pnorm(q = 20, mean = 30.2, sd = 13.8) # paste R outputs here 0.2299139 What is the probability that a randomly selected mother in Yuma County gained between 25 to 35 pounds (i.e., within the recommended range of weight gain)? [2 points] Approach A (compute manually i.e., without R) 𝑍 = 𝑥 − ? 𝜎 𝑍 = 25−30.2 13.8 = −0.38 (z-table value is 0.3520); 𝑍 = 35−30.2 13.8 = 0.35 (z-table value is 0.6368) 0.6368 − 0.3520 = 0.2848 [1 point] Approach B (use R) # include R commands here > small <- pnorm(q = 25, mean = 30.2, sd = 13.8) > large <- pnorm(q = 35, mean = 30.2, sd = 13.8) > large - small # paste R outputs here 0.282858 What is the probability that a randomly selected mother in Yuma County gained more than 40 pounds (i.e., more the recommended amount of weight gain)? [2 points] Approach A (compute manually i.e., without R) 𝑍 = 𝑥 − ? 𝜎

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

𝑍 = 40 − 30.2 13.8 = 0.710 Using the positive Z-table, the probability with the above z-score is 0.7611, but you must subtract that from 1 which gives a probability of: 1 − 0.7611 = 0.2389 [1 point] Approach B (use R) # include R commands here > 1 - pnorm(q = 40, mean = 30.2, sd = 13.8) # paste R outputs here 0.2388071 4. [5 points] In a Southern Arizona border town, an average of 14 new cases of esophageal cancer are screened and diagnosed each year. Assuming that the incidence of esophageal cancer follows a Poisson distribution, what is the probability that in a given year the number of newly diagnosed cases of esophageal cancer will be less than 10? [1 point] Use R # include R commands here > ppois(9, lambda = 14) # paste R outputs here 0.1093994 What is the probability that in a given year the number of newly diagnosed cases of esophageal cancer will be between 10 and 20 (inclusive)? [1 point] Use R # include R commands here > dpois(10, lambda = 14) + dpois(11, lambda = 14) + dpois(12, lambda = 14) + dpois(13, lambda = 14) + dpois(14, lambda = 14) + dpois(15, lambda = 14) + dpois(16, lambda = 14) + dpois(17, lambda = 14) + dpois(18, lambda = 14) + dpois(19, lambda = 14) + dpois(20, lambda = 14) # paste R outputs here 0.8426922 What is the probability that in a given year the number of newly diagnosed cases of esophageal cancer will be exactly 10? [2 points] Approach A (compute manually i.e., without R) 𝑃(𝑥 = 𝐾) = 𝑒 −𝜆 ∗ ? 𝑘 𝑘! 𝑃(𝑥 = 10) = 𝑒 −14 ∗ 14 10 10! = 0.066282 [1 point] Approach B (use R) # include R commands here > dpois(10, lambda = 14) # paste R outputs here 0.06628184

5. [1 point] Statement of Attribution and Challenges: Please list and describe any resource or external support used to complete this assignment. You should clearly specify and attribute any source of support, including but not limited to, the use of online and physical materials, software tools and websites, peers, human tutors, digital tutors, programming assistants, and any other form of artificial intelligence (AI) tools. You are also welcome to include questions or general comments/challenges that came up while you were doing this assignment. I used the online notes that were provided and discussed in class. I also did ask a couple of classmates for help on some of the problems. The problems that involved ranges I asked Julia for help on and some of the manual calculations I got help from Ryan on.