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	15.8	19.3	14.4	19.6	20.0

Σx = 15; Σy = 89.1; Σx² = 55; Σy² = 1613.65; Σxy = 276

(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.     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.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.

(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 > 0.2500.125 < P-value < 0.250     0.100 < P-value < 0.1250.075 < P-value < 0.1000.050 < P-value < 0.0750.025 < P-value < 0.0500.010 < P-value < 0.0250.005 < P-value < 0.0100.0005 < P-value < 0.005P-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 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.     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.

Answer 1

We have Σx = 15; Σy = 89.1; Σx² = 55; Σy² = 1613.65; Σxy = 276, N=5

Our aim is to calculate the values m (slope) and b (y-intercept) in the equation of the least-squares Line.

y = mx + b

Calculate Slope m:

m = N Σ(xy) − Σx Σy / N Σ(x²) − (Σx)²

   =5*276-15*89.1/5*55-15²

=45/50 =0.87

Calculate Intercept b:

b = Σy − m Σx / N

=89.1-(0.87*15)/5

=17.28

Assemble the equation of a line:

y = mx + b

y =0.87*x+17.28

Part c.

  

    

  

So

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.    

Part d.

• Null Hypothesis: H₀: ρ = 0
• Alternate Hypothesis: H_a: ρ > 0

  

  

n – 2 degrees of freedom.=5-2=3

Using Excel ''=TDIST(1.114,3,1)'' we will get 0.1732

P-value > 0.01 so we do not  reject the null hypothesis.

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

Hope this will be helpful. Thanks and God Bless You:)