Question

5. The data below represent duration times in seconds) of eruptions and time intervals (in minutes) to the next eruption for randomly selected eruptions of the Old Faithful geyser in Yellowstone National Park. Duration 242 255 227 251 262 207140 Interval After 91 81 91 92 102 94 91 a. Find the regression equation. b. Construct a residual plot for the data. Use the table below to guide you. y-9 Point on Plot Y c. Is a linear model appropriate for this data? Explain your answer.

Answer 1

Solution:

I have used excel to solve this problem.

a) Let 'x' be the independent variable 'duration' and 'y' be the dependent variable 'interval after'.

Duration(x)	Interval After(y)
242	91
255	81
227	91
251	92
262	102
207	94
140	91

In excel go to Data-->Data analysis-->Regression-->Ok.

The below is the screenshot for your reference.

FILE HOME INSERT PAGE LAYOUT FORMULAS DATA Developer REVIEW VIEW ADD-INS LOAD TEST POWERPIVOT TEAM Si Connections Clear At A4 + Data Analysis tot En From Access Le From Web From Other Lh From Text Sources Get External Data Sort Filter Existing Connections Properties Refresh All Edit Links Connections Reapply Advanced Text to Columns Flash Remove Data Consolidate What If Relationships Group Ungroup Subtotal Fill Duplicates Validation Analysis Data Tools Outline Sort & Filter Analysis E12 х f A с D E F H - J K L M N o P Q R 1 Duration(x) 242 2 B Interval After(y) 91 81 91 92 3 255 4 227 5 251 6 262 102 94 7 207 140 91 8 9 ? х OK 10 SUMMARY OUTPUT 11 12 Data Analysis 13 Analysis Tools 14 F-Test Two-Sample for Variances 15 Fourier Analysis Histogram 16 Moving Average Random Number Generation 17 Rank and Percentile 18 Regression Sampling 19 -Test: Paired Two Sample for Means Cancel Help

• Select Interval After as 'Input Y Range' and Duration as 'Input X Range'.
• Click on Label check box since first row has labels.
• Select any range of excel sheet as output range.Click ok.

The below is the screenshot for your reference.

B1 fx с D E F H I B Interval After(y): 91 IU 3 A Duration(x) 242 255 227 251 262 81 1 ? x 4 91 Regression - UN UI 92 102 Input Input Y Range: OK SBS1:5858 94 Cancel 207 140 Input X Range: 91 SA$1:$A$8 Help 0 SUMMARY OUTPUT Labels Confidence Level: Constant is Zero 95 % N SA$10 Time 1 Regression Statistics 3 Multiple R 0.046358127 4 R Square 0.002149076 5 Adjusted R Square -0.197421109 6 Standard Error 6.737058856 7 Observations 7 Output options Output Range O New Worksheet Ply: New Workbook Residuals Residuals Standardized Residuals Residual Plots Line Fit Plots 8 Normal Probability Normal Probability Plots df 9 ANOVA 0 1 Regression 2 Residual 3 Total 5 226.9398 45.38796 6 227.4286

The regression equation for the given data is as follows:

	Coefficients
Intercept	90.1903
Duration(x)	0.0067

The general form of regression equation is$\hat{y}$=ax+b where a=slope and b=intercept

So $\hat{y}$ =0.0067x+90.1903

b)Residual plots for the data is as follows:

Duration(x)	Interval After(y)	$\hat{y}$	y-$\hat{y}$	Point on plot(x,y-$\hat{y}$)
242	91	91.81	91-91.81=-0.81	(242,-0.81)
255	81	91.90	81-91.90=-10.90	(255,-10.90)
227	91	91.71	91-91.71=-0.71	(227,-0.71)
251	92	91.87	92-91.87=0.13	(251,0.13)
262	102	91.95	102-91.95=10.05	(262,10.05)
207	94	91.58	94-91.58=2.42	(207,2.42)
140	91	91.13	91-91.13=-0.13	(140,-0.13)

The graph of residual plot is as follows:

Residual Plot 10 Residuals 0 50 100 150 200 250 300 -10 -20 Duration(x)

c) The residual plot in the above graph shows a fairly random pattern. The first three residual is negative,the next three are positive and the last is negative. This random pattern indicates that a linear model provides a decent fit to the data.