listed below are altitudes (thousands of feet) and outside air temperatures (°f) recorded during a flight…

listed below are altitudes (thousands of feet) and outside air temperatures (°f) recorded during a flight. find the (a) explained variation, (b) unexplained variation, and (c) indicated prediction interval. there is sufficient evidence to support a claim of a linear correlation, so it is reasonable to use the regression equation when making predictions. for the prediction interval, use a 95% confidence level with the altitude of 6327 ft (or 6.327 thousand feet).\naltitude | 4 | 12 | 16 | 23 | 27 | 31 | 34\ntemperature | 60 | 40 | 27 | 1 | - 30 | - 41 | - 55\na. find the explained variation.\n11,280.01\n(round to two decimal places as needed.)\nb. find the unexplained variation.\n(round to five decimal places as needed.)

listed below are altitudes (thousands of feet) and outside air temperatures (°f) recorded during a flight. find the (a) explained variation, (b) unexplained variation, and (c) indicated prediction interval. there is sufficient evidence to support a claim of a linear correlation, so it is reasonable to use the regression equation when making predictions. for the prediction interval, use a 95% confidence level with the altitude of 6327 ft (or 6.327 thousand feet).\naltitude | 4 | 12 | 16 | 23 | 27 | 31 | 34\ntemperature | 60 | 40 | 27 | 1 | - 30 | - 41 | - 55\na. find the explained variation.\n11,280.01\n(round to two decimal places as needed.)\nb. find the unexplained variation.\n(round to five decimal places as needed.)

Answer

Explanation:

Step1: Calculate necessary sums

Let (x) be altitude and (y) be temperature. First, find (\sum x), (\sum y), (\sum x^2), (\sum y^2), (\sum xy), (n = 7).

Step2: Find the regression equation (\hat{y}=a + bx)

Use the formulas (b=\frac{n\sum xy-\sum x\sum y}{n\sum x^2 - (\sum x)^2}) and (a=\bar{y}-b\bar{x}), where (\bar{x}=\frac{\sum x}{n}) and (\bar{y}=\frac{\sum y}{n}).

Step3: Calculate total variation (SST=\sum(y - \bar{y})^2)

(SST=\sum y^2-\frac{(\sum y)^2}{n})

Step4: Calculate explained variation (SSR)

(SSR = b^2\left(\sum x^2-\frac{(\sum x)^2}{n}\right)) (already given as (11280.01))

Step5: Calculate unexplained variation (SSE)

(SSE=SST - SSR)

Let's assume we have calculated the following sums: (\sum x=4 + 12+16+23+27+31+34 = 147) (\sum y=60 + 40+27+1+( - 30)+( - 41)+( - 55)=0) (\sum x^2=4^2 + 12^2+16^2+23^2+27^2+31^2+34^2=3939) (\sum y^2=60^2 + 40^2+27^2+1^2+( - 30)^2+( - 41)^2+( - 55)^2=10166) (\sum xy=4\times60+12\times40 + 16\times27+23\times1+27\times( - 30)+31\times( - 41)+34\times( - 55)=- 2914)

(\bar{x}=\frac{147}{7}=21) (\bar{y}=\frac{0}{7}=0)

(b=\frac{7\times(- 2914)-147\times0}{7\times3939 - 147^2}=\frac{-20398}{27573 - 21609}=\frac{-20398}{5964}\approx - 3.42) (a = 0-(-3.42)\times21 = 71.82)

(SST=10166-\frac{0^2}{7}=10166) (SSR = 11280.01) (SSE=10166 - 11280.01=-1114.01) (There is a mistake above, we should use the correct formula (SSE=\sum(y - \hat{y})^2))

First, for each (x_i), calculate (\hat{y}_i=a + bx_i) For (x_1 = 4), (\hat{y}_1=71.82-3.42\times4=71.82 - 13.68 = 58.14) ((y_1-\hat{y}_1)^2=(60 - 58.14)^2 = 3.4596)

After calculating ((y_i-\hat{y}_i)^2) for all (i) and summing them up: Let (y) values be (y_1 = 60,y_2 = 40,y_3 = 27,y_4 = 1,y_5=-30,y_6=-41,y_7=-55) (\hat{y}_2=71.82-3.42\times12=71.82-41.04 = 30.78), ((y_2 - \hat{y}_2)^2=(40 - 30.78)^2=85.0084) (\hat{y}_3=71.82-3.42\times16=71.82 - 54.72=17.1), ((y_3 - \hat{y}_3)^2=(27 - 17.1)^2 = 98.01) (\hat{y}_4=71.82-3.42\times23=71.82-78.66=-6.84), ((y_4 - \hat{y}_4)^2=(1+6.84)^2 = 61.4656) (\hat{y}_5=71.82-3.42\times27=71.82 - 92.34=-20.52), ((y_5 - \hat{y}_5)^2=(-30 + 20.52)^2=89.8704) (\hat{y}_6=71.82-3.42\times31=71.82-106.02=-34.2), ((y_6 - \hat{y}_6)^2=(-41 + 34.2)^2 = 46.24) (\hat{y}_7=71.82-3.42\times34=71.82-116.28=-44.46), ((y_7 - \hat{y}_7)^2=(-55 + 44.46)^2 = 111.5916)

(SSE=3.4596 + 85.0084+98.01+61.4656+89.8704+46.24+111.5916 = 495.646)

Answer:

b. (495.64600)