University:
Washington State University
Course:
MATH 172 | Calculus II
Academic year:

2024
Views:

151

Pages:

3
Author:

Bailey Ayala

Question 1 Why use least-squares lines for one scatterplot but not the other? Answer: Scatterplot 1: The points are scattered without any discernible linear pattern. A least-squares regression line is unsuitable as there is no linear relationship between 𝑥 x and 𝑦 y. Scatterplot 2: The points form a trend where 𝑦 y increases with 𝑥 x, then decreases. A least-squares regression line can approximate this relationship since the pattern suggests some form of linearity or trend between 𝑥 x and 𝑦 y. Question 2 Matching scatterplots to correlations (-0.923, -0.487, 0.006, 0.777): Answer: Scatterplot (a): 0.777 0.777 (Strong positive correlation, upward trend). Scatterplot (b): − 0.487 −0.487 (Moderate negative correlation, downward trend with scatter). Scatterplot (c): 0.006 0.006 (No correlation, points are scattered randomly). Scatterplot (d): − 0.923 −0.923 (Strong negative correlation, steep downward trend). Question 3 An association is nonlinear when there is curvature in the scatterplot. The correlation is close to 𝑟 = − 1 r=−1. Answer: A nonlinear association shows a curved pattern in the scatterplot. When 𝑟 = − 1 r=−1, the data points closely follow a strong downward trend, but nonlinearity introduces curvature while maintaining a strong correlation. For example, a downward curve with minimal deviation still has 𝑟 ≈ − 1 r≈−1. Question 4 Percent of 15-year-old students taller than 180 cm: Given: Regression equation: 𝜇 𝑦 = 105 + 4.2 𝑥 μ y =105+4.2x, where 𝑥 = age in years x=age in years, 𝜎 = 7 σ=7. Age 𝑥 = 15 x=15. Calculate the mean height for 15-year-olds ( 𝜇 𝑦 μ y ): 𝜇 𝑦 = 105 + 4.2 ( 15 ) = 105 + 63 = 168 cm μ y =105+4.2(15)=105+63=168cm Find the Z-score for 𝑋 = 180 X=180: 𝑧 = 𝑥 − 𝜇 𝑦 𝜎 = 180 − 168 7 = 12 7 ≈ 1.71 z= σ x−μ y = 7 180−168 = 7 12 ≈1.71 Find 𝑃 ( 𝑍 > 1.71 ) P(Z>1.71): From standard normal tables: 𝑃 ( 𝑍 < 1.71 ) = 0.9564 , 𝑃 ( 𝑍 > 1.71 ) = 1 − 0.9564 = 0.0436 P(Z<1.71)=0.9564,P(Z>1.71)=1−0.9564=0.0436 Answer: Approximately 4.36% of 15-year-old students are expected to be taller than 180 cm.