Question

Excerpts from Elements of Ecology 3^rd edition Robert Leo Smith: “The critical time of year for the African buffalo is the dry season, when food becomes scarce. Rainfall determines the productivity of grass. The greater the rainfall, the more vigorously the grass grows, increasing the amount of forage available in the dry season…. the greater the rainfall, the greater density of buffalo”

Annual Rainfall (cm)	64	72	82	88	113	120	150	155
Buffalo Population (per sq km.)	1	3	4	13	7	11	20	15

1. Compute the correlation coefficient between rainfall and buffalo population

R=.8621

2. Test, at  whether the linear relation between annual rainfall and population density of the African buffalo is statistically significant. Be sure to include , , the test statistic and its p value, and your conclusion.

Ho: p=0

Ha: p≠0

T=4.17

P=.0059

Reject Ho, there is sufficient evidence to show the relationship between annual rainfall and population density of the African buffalo are correlated.

c). Provide the regression line

y=-8.095+.164x

d). If forecasters predict that in one region that African buffaloes inhabit annual rainfall will be 100cm, estimate the population density of African buffaloes.

e). If, in another region in which African buffaloes inhabit, forecasters predict that over time annual rainfall will drop by 1 cm, what do you estimate the consequence on the African buffaloes population density to be?

Answer 1

The regression output is as follows

Question (a)

The correlation coefficient between rainfall and buffalo population is the Mutiple R value from the Regression statistics section in the image above

So correlation coefficient r = 0.8621

Question (b)

To test whether the linear relation between annual rainfall and population density of the African buffalo is statistically significant or not we need to test the hypothesis on slope of the regression line 1

H0: 1 =0

Ha: 1 ≠ 0

The test-statistic value is the t Stat value of Annual Rainfall in the attached image which is 4.1669

So test-statistic = 4.1669

The p-value is the P-value of Annual Rainfall in the attached image which is 0.0059

So P-value = 0.0059

Since the p-value is less than the significance level of 0.05, we Reject H0 and conclude that there is sufficient evidence to show the relationship between annual rainfall and population density of the African buffalo are correlated

Question (c)

The equation of regressio line is

y = -8.0947 + 0.1644* x

Where y is the population density of the African buffalo

x is the Annual rainfall

Question (d)

Given that in one region that African buffaloes inhabit annual rainfall will be 100cm

so y = -8.0947 + 0.1644* 100

= -8.0947 + 16.44

= 8.3453

So the population density of African buffaloes is 8.3453 for a annual rainfall of 100cm

Question (e)

Over time annual rainfall will drop by 1 cm

If the annual rainfall drops by 1cm, then the African buffaloes population density will drop by 0.1644 becuase the coefficient of Annual rainfall is 0.1644

Both Annual rainfall and African buffaloes population density are positively correlated which implies, if the Annual rainfall increased by 1 unit, then the African buffaloes population density increases by 0.1644 units and if the Annual rainfall decreased by 1 unit, then the African buffaloes population density decreases by 0.1644 units

So Answer is that the African buffaloes population density will drop by 0.1644 per sq.km

Anuual Rainfall 64 Buffalo Population 1 SUMMARY OUTPUT 72 3 4 82 88 113 13 7 Regression Statistics Multiple R 0.8621 R Square 0.7432 Adjusted R Square 0.7004 Standard Error 3.6161 Observations 8 120 11 150 20 15 155 ANOVA ss MS F Significance F 0.0059 227.0430 Regression Residual Total df 1 6 7 227.0430 17.3631 13.0762 78.4570 305.5000 t Stat Intercept Annual Rainfall Coefficients Standard Error -8.0947 4.3544 0.1644 0.0395 -1.8590 4.1669 P-value 0.1124 0.0059 Lower 95% - 18.7495 0.0679 Upper 95% 2.5601 0.2609

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