Question

Bighorn sheep are beautiful wild animals found throughout the western United States. Let x be the age of a bighorn sheep (in years), and let y be the mortality rate (percent that die) for this age group. For example, x = 1, y = 14 means that 14% of the bighorn sheep between 1 and 2 years old died. A random sample of Arizona bighorn sheep gave the following information:

x	1	2	3	4	5
y	12.8	20.9	14.4	19.6	20.0

Σx = 15; Σy = 87.7; Σx² = 55; Σy² = 1592.17; Σxy = 276.2

(a) Draw a scatter diagram.

(b) Find the equation of the least-squares line. (Round your answers to two decimal places.)

ŷ =

+   x

(c) Find r. Find the coefficient of determination r². (Round your answers to three decimal places.)

r =
r2 =

Explain what these measures mean in the context of the problem.

The correlation coefficient r measures the strength of the linear relationship between a bighorn sheep's age and the mortality rate. The coefficient of determination r² measures the explained variation in mortality rate by the corresponding variation in age of a bighorn sheep.

The coefficient of determination r measures the strength of the linear relationship between a bighorn sheep's age and the mortality rate. The correlation coefficient r² measures the explained variation in mortality rate by the corresponding variation in age of a bighorn sheep.  

Both the correlation coefficient r and coefficient of determination r² measure the strength of the linear relationship between a bighorn sheep's age and the mortality rate.

The correlation coefficient r² measures the strength of the linear relationship between a bighorn sheep's age and the mortality rate. The coefficient of determination r measures the explained variation in mortality rate by the corresponding variation in age of a bighorn sheep.

(d) Test the claim that the population correlation coefficient is positive at the 1% level of significance. (Round your test statistic to three decimal places.)

t =

Find or estimate the P-value of the test statistic.

P-value > 0.250

0.125 < P-value < 0.250   

0.100 < P-value < 0.125

0.075 < P-value < 0.100

0.050 < P-value < 0.075

0.025 < P-value < 0.050

0.010 < P-value < 0.025

0.005 < P-value < 0.010

0.0005 < P-value < 0.005

P-value < 0.0005

Conclusion

Reject the null hypothesis, there is sufficient evidence that ρ > 0.

Reject the null hypothesis, there is insufficient evidence that ρ > 0.   

Fail to reject the null hypothesis, there is sufficient evidence that ρ > 0.

Fail to reject the null hypothesis, there is insufficient evidence that ρ > 0.

(e) Given the result from part (c), is it practical to find estimates of y for a given x value based on the least-squares line model? Explain.

Given the lack of significance of r, prediction from the least-squares model might be misleading.

Given the significance of r, prediction from the least-squares model is practical.  

Given the significance of r, prediction from the least-squares model might be misleading.

Given the lack of significance of r, prediction from the least-squares model is practical.

Answer 1

from above:

a) scatter diagram is attached above

b)

equation of the least-squares line :y^ =13.61+1.31x

c)r =0.564

r² =0.318

The correlation coefficient r measures the strength of the linear relationship between a bighorn sheep's age and the mortality rate. The coefficient of determination r² measures the explained variation in mortality rate by the corresponding variation in age of a bighorn sheep.

d)

t =1.184

0.125 < P-value < 0.250

Fail to reject the null hypothesis, there is insufficient evidence that ρ > 0.

e)

Given the lack of significance of r, prediction from the least-squares model might be misleading.