unit 1 do nows\ndo now 1 - 1.1 function basics\n1) in which quadrant do lines l and m intersect?\nname…

unit 1 do nows\ndo now 1 - 1.1 function basics\n1) in which quadrant do lines l and m intersect?\nname brianna moore\n1. the data in the table below shows the average temperature in northern latitudes:\n| latitude (°n) | 0 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 |\n| temp (°f) | 79 | 81 | 79 | 68 | 58 | 43 | 28 | 13 | 1 |\n a) find the line of best fit: \n b) estimate the average temperature for a city with a latitude of 48°: \n2. the data in the table below shows the number of passengers and number of suitcases on various airplanes.\n| passengers | 75 | 92 | 115 | 128 | 143 | 154 | 178 | 200 |\n| suitcases | 159 | 180 | 239 | 272 | 290 | 310 | 357 | 405 |\n a) find the line of best fit: \n b) estimate the number of suitcases on a flight carrying 250 people.\n3. the data in the table below shows the number of graduating seniors at canyon valley high school since 2012.\n| year | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 |\n| graduates | 340 | 348 | 356 | 361 | 375 | 387 |\n a) find the line of best fit: \n b) estimate the number of graduating seniors in 2025.\n4. the data in the table to the left shows the olympic 500 - meter gold medal speed skating times.\n| year | time (s) |\n| 1980 | 422 |\n| 1984 | 432 |\n| 1988 | 404 |\n| 1992 | 420 |\n| 1994 | 395 |\n| 1998 | 382 |\n a) find the line of best fit: \n b) estimate the 500 - meter time for the 2020 olympics.\n c) write a description of data: \n c) write a description of data: \n c) write a description of data: \n c) write a description of data:

unit 1 do nows\ndo now 1 - 1.1 function basics\n1) in which quadrant do lines l and m intersect?\nname brianna moore\n1. the data in the table below shows the average temperature in northern latitudes:\n| latitude (°n) | 0 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 |\n| temp (°f) | 79 | 81 | 79 | 68 | 58 | 43 | 28 | 13 | 1 |\n a) find the line of best fit: \n b) estimate the average temperature for a city with a latitude of 48°: \n2. the data in the table below shows the number of passengers and number of suitcases on various airplanes.\n| passengers | 75 | 92 | 115 | 128 | 143 | 154 | 178 | 200 |\n| suitcases | 159 | 180 | 239 | 272 | 290 | 310 | 357 | 405 |\n a) find the line of best fit: \n b) estimate the number of suitcases on a flight carrying 250 people.\n3. the data in the table below shows the number of graduating seniors at canyon valley high school since 2012.\n| year | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 |\n| graduates | 340 | 348 | 356 | 361 | 375 | 387 |\n a) find the line of best fit: \n b) estimate the number of graduating seniors in 2025.\n4. the data in the table to the left shows the olympic 500 - meter gold medal speed skating times.\n| year | time (s) |\n| 1980 | 422 |\n| 1984 | 432 |\n| 1988 | 404 |\n| 1992 | 420 |\n| 1994 | 395 |\n| 1998 | 382 |\n a) find the line of best fit: \n b) estimate the 500 - meter time for the 2020 olympics.\n c) write a description of data: \n c) write a description of data: \n c) write a description of data: \n c) write a description of data:

Answer

Explanation:

Step1: Recall the formula for the line of best - fit (least - squares regression line)

The line of best - fit for a set of data points ((x_i,y_i)) is of the form (y = mx + b), where (m=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^{2}-(\sum_{i = 1}^{n}x_i)^{2}}) and (b=\overline{y}-m\overline{x}), (\overline{x}=\frac{1}{n}\sum_{i = 1}^{n}x_i), (\overline{y}=\frac{1}{n}\sum_{i = 1}^{n}y_i), and (n) is the number of data points.

Step2: For the first data set (latitude - temperature)

Let (x) be the latitude and (y) be the temperature.

  1. Calculate (n = 9), (\sum_{i=1}^{9}x_i=0 + 10+20+\cdots+80 = 360), (\sum_{i = 1}^{9}y_i=79 + 81+\cdots+1=392), (\sum_{i=1}^{9}x_i^{2}=0^{2}+10^{2}+\cdots+80^{2}=20400), (\sum_{i = 1}^{9}x_iy_i=0\times79+10\times81+\cdots+80\times1 = 1470).
  2. Calculate (m=\frac{9\times1470 - 360\times392}{9\times20400-(360)^{2}}) [ \begin{align*} m&=\frac{13230-141120}{183600 - 129600}\ &=\frac{- 127890}{54000}\ &=- 2.3683 \end{align*} ]
  3. Calculate (\overline{x}=\frac{360}{9}=40), (\overline{y}=\frac{392}{9}\approx43.56)
  4. Calculate (b = 43.56-(-2.3683)\times40=43.56 + 94.732=138.292) The line of best - fit is (y=-2.3683x + 138.292)
  5. For (x = 48), (y=-2.3683\times48+138.292=-113.6784 + 138.292 = 24.6136\approx24.61)

Step3: For the second data set (passengers - suitcases)

Let (x) be the number of passengers and (y) be the number of suitcases.

  1. Calculate (n = 8), (\sum_{i = 1}^{8}x_i=75 + 92+\cdots+200 = 1085), (\sum_{i = 1}^{8}y_i=159+180+\cdots+405 = 1922), (\sum_{i = 1}^{8}x_i^{2}=75^{2}+92^{2}+\cdots+200^{2}=177933), (\sum_{i = 1}^{8}x_iy_i=75\times159+92\times180+\cdots+200\times405 = 303995)
  2. Calculate (m=\frac{8\times303995-1085\times1922}{8\times177933-(1085)^{2}}) [ \begin{align*} m&=\frac{2431960 - 2085370}{1423464-1177225}\ &=\frac{346590}{246239}\ &\approx1.4075 \end{align*} ]
  3. Calculate (\overline{x}=\frac{1085}{8}=135.625), (\overline{y}=\frac{1922}{8}=240.25)
  4. Calculate (b = 240.25-1.4075\times135.625=240.25 - 190.877=49.373) The line of best - fit is (y = 1.4075x+49.373)
  5. For (x = 250), (y=1.4075\times250+49.373=351.875+49.373 = 401.248\approx401.25)

Step4: For the third data set (year - graduates)

Let (x) be the number of years since 2012 ((x = 0) for 2012, (x = 1) for 2013, etc.).

  1. Calculate (n = 6), (\sum_{i = 1}^{6}x_i=0 + 1+\cdots+5 = 15), (\sum_{i = 1}^{6}y_i=340+348+\cdots+387 = 2167), (\sum_{i = 1}^{6}x_i^{2}=0^{2}+1^{2}+\cdots+5^{2}=55), (\sum_{i = 1}^{6}x_iy_i=0\times340+1\times348+\cdots+5\times387 = 5915)
  2. Calculate (m=\frac{6\times5915-15\times2167}{6\times55-(15)^{2}}) [ \begin{align*} m&=\frac{35490-32505}{330 - 225}\ &=\frac{2985}{105}\ &=28.4286 \end{align*} ]
  3. Calculate (\overline{x}=\frac{15}{6}=2.5), (\overline{y}=\frac{2167}{6}\approx361.17)
  4. Calculate (b = 361.17-28.4286\times2.5=361.17 - 71.0715=290.0985) The line of best - fit is (y = 28.4286x+290.0985)
  5. For (x = 13) (2025 - 2012), (y=28.4286\times13+290.0985=369.5718+290.0985 = 659.6703\approx659.67)

Step5: For the fourth data set (year - time)

Let (x) be the number of years since 1980 ((x = 0) for 1980, (x = 4) for 1984, etc.).

  1. Calculate (n = 6), (\sum_{i = 1}^{6}x_i=0 + 4+\cdots+18 = 44), (\sum_{i = 1}^{6}y_i=422+432+\cdots+382 = 2455), (\sum_{i = 1}^{6}x_i^{2}=0^{2}+4^{2}+\cdots+18^{2}=580), (\sum_{i = 1}^{6}x_iy_i=0\times422+4\times432+\cdots+18\times382 = 16778)
  2. Calculate (m=\frac{6\times16778-44\times2455}{6\times580-(44)^{2}}) [ \begin{align*} m&=\frac{100668-107020}{3480 - 1936}\ &=\frac{- 6352}{1544}\ &=- 4.114 \end{align*} ]
  3. Calculate (\overline{x}=\frac{44}{6}\approx7.33), (\overline{y}=\frac{2455}{6}\approx409.17)
  4. Calculate (b = 409.17-(-4.114)\times7.33=409.17+30.1556 = 439.3256) The line of best - fit is (y=-4.114x + 439.3256)
  5. For (x = 40) (2020 - 1980), (y=-4.114\times40+439.3256=-164.56+439.3256 = 274.7656\approx274.77)

Answer:

a) (y=-2.3683x + 138.292) b) (24.61) c) As the latitude increases, the average temperature decreases approximately linearly. 2. a) (y = 1.4075x+49.373) b) (401.25) c) The number of suitcases increases approximately linearly with the number of passengers. 3. a) (y = 28.4286x+290.0985) b) (659.67) c) The number of graduating seniors has been increasing approximately linearly since 2012. 4. a) (y=-4.114x + 439.3256) b) (274.77) c) The Olympic 500 - meter Gold Medal Speed Skating times have been decreasing approximately linearly over the years.