Question

Cameron is investigating how long his phone's battery lasts (in hours) for various brightness levels (on a scale of O- 100). His data is displayed in the table and graph below. Brightness Level (x) Hours (y) 16 28 31 43 49 73 86 5 0.1 6.6 5.1 5.2 6 .5 10+ 9 8 7 . 6 . 5 . . Hours w 2 1 40 60 70 84 10 20 30 50 Brightness Level a) Find the equation for the line of best fit. Keep at least 4 decimals for each parameter in the equation. b) Interpret the slope in context. Cameron should expect -0.0636 hours per brightness level. Cameron should expect -0.0636 brightness level per hour. c) What does the equation predict for the number of hours the phone will last at a brightness level of 73? hours d) What is the residual for the point (73,5)? hours

Answer 1

Brightness Level (X)	Hours (y)
16	6.6
28	5.1
31	5.2
43	6
49	4.5
73	5
86	0.1
88	2.3

Steps:

Enter data > insert > scatterplot > add trendline and equation > ok

Scatterplot with equation 7 6 5 Hours (v) 2 y = -0.0636x + 7.6414 R2 = 0.674 0 0 20 40 60 80 100 Brightness Level(x)

steps:

Enter data > Data > analyse > data analysis > regression > input x and y range > ok

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.820997151
R Square	0.674036322	It explains 67.4 % variability in y
Adjusted R Square	0.619709043
Standard Error	1.31397144
Observations	8
ANOVA
	df	SS	MS	F	Significance F
Regression	1	21.42087433	21.42087433	12.40695885	0.012482904	significant at 0.05
Residual	6	10.35912567	1.726520945
Total	7	31.78
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	7.641360589	1.043531764	7.32259511	0.000331308	5.08793035	10.19479083
Brightness Level (X)	-0.063601171	0.018056453	-3.522351324	0.012482904	-0.107783719	-0.019418622

Equation:

Hours = 7.6414 -0.0636 Brightness Level

Now, interpretation of slope:

For a 100% increase in Brightness Level, the hours decrease by 6.36%

For a brightness = 73

Hours = 7.6414 - 0.06363 * 73 = 2.99641 or 3 hours

Error / residual:

Actual value at x = 73, y = 5 hours

Predicted value at x = 73 , $\hat y = 3$

Error = $\hat y - y$ = 3 - 5 = -2 hours

Note: sign of the residual doesnot matter because the mean is always 0.

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