Nichols-Assignment-D

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University Of Arizona *

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376

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Mechanical Engineering

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Jan 9, 2024

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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

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