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 | 7 | 14 | 21 | 27 | 31 | 33\ntemperature | 55 | 32 | 28 | - 4 | - 32 | - 41 | - 58\na. find the explained variation.\n10,556.70\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 | 7 | 14 | 21 | 27 | 31 | 33\ntemperature | 55 | 32 | 28 | - 4 | - 32 | - 41 | - 58\na. find the explained variation.\n10,556.70\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 the regression equation

First, we need to calculate the means of altitude ($\bar{x}$) and temperature ($\bar{y}$), the sums of squares $SS_{xx}$, $SS_{yy}$, $SS_{xy}$, the slope $b_1=\frac{SS_{xy}}{SS_{xx}}$ and the y - intercept $b_0=\bar{y}-b_1\bar{x}$ of the regression line $\hat{y}=b_0 + b_1x$. Let $x$ be the altitude and $y$ be the temperature. $n = 7$ $\sum x=4 + 7+14+21+27+31+33=137$ $\sum y=55 + 32+28-4-32-41-58= - 20$ $\bar{x}=\frac{\sum x}{n}=\frac{137}{7}\approx19.57143$ $\bar{y}=\frac{\sum y}{n}=\frac{-20}{7}\approx - 2.85714$ $SS_{xx}=\sum x^{2}-\frac{(\sum x)^{2}}{n}$ $\sum x^{2}=4^{2}+7^{2}+14^{2}+21^{2}+27^{2}+31^{2}+33^{2}=16 + 49+196+441+729+961+1089 = 3481$ $SS_{xx}=3481-\frac{137^{2}}{7}=3481-\frac{18769}{7}=3481 - 2681.28571=799.71429$ $SS_{yy}=\sum y^{2}-\frac{(\sum y)^{2}}{n}$ $\sum y^{2}=55^{2}+32^{2}+28^{2}+(-4)^{2}+(-32)^{2}+(-41)^{2}+(-58)^{2}=3025+1024+784 + 16+1024+1681+3364=10918$ $SS_{yy}=10918-\frac{(-20)^{2}}{7}=10918-\frac{400}{7}=10918 - 57.14286=10860.85714$ $SS_{xy}=\sum xy-\frac{\sum x\sum y}{n}$ $\sum xy=(4\times55)+(7\times32)+(14\times28)+(21\times(-4))+(27\times(-32))+(31\times(-41))+(33\times(-58))$ $=220+224+392-84 - 864-1271-1914=-3297$ $SS_{xy}=-3297-\frac{137\times(-20)}{7}=-3297+\frac{2740}{7}=-3297 + 391.42857=-2905.57143$ $b_1=\frac{SS_{xy}}{SS_{xx}}=\frac{-2905.57143}{799.71429}\approx - 3.63333$ $b_0=\bar{y}-b_1\bar{x}=-2.85714-(-3.63333)\times19.57143=-2.85714 + 70.19048 = 67.33334$ The regression equation is $\hat{y}=67.33334-3.63333x$

Step2: Calculate the total variation $SS_{yy}$

We have already calculated $SS_{yy}=10860.85714$

Step3: Calculate the explained variation

The explained variation is $SSR = b_1^{2}SS_{xx}$ $SSR=(-3.63333)^{2}\times799.71429$ $SSR = 13.19047\times799.71429\approx10556.70$ (already given in the problem - used for verification)

Step4: Calculate the unexplained variation

The unexplained variation is $SSE=SS_{yy}-SSR$ $SSE = 10860.85714-10556.70=304.15714$

Answer:

304.15714