Question

7. Psychological and social factors can influence the survival of patients with serious diseases. One study compared the relationship between survival of patients with coronary heart disease and pet ownership. Patients were either classified as having a pet or not and whether they survived for 1 year. Based on the data below, is there an association between owning a pet and survival? No need to do the entire significance test, just the parts below. Pet Ownership Patient Status Yes Alive Dead 3 a. State the hypotheses for the appropriate test in the context of this problem. No 28 50 b. If the X test statistic has a value of 8.85 then give the p-value and conclusion of your significance test 8. A researcher wishes to determine whether the rate of water flow (x) (in liters per second) over an experimental soil bed can be used to predict the amount of soil washed away(y) (in kilograms). The researcher measures the amount of soil washed away for various flow rates, finds that the sample mean flow rate is 6.1 L's with a standard deviation of 0.5 L/s. Also the sample mean for the amount of eroded soil is 4.2 kg with a standard deviation of 1.2 kg. If the correlation between flow rate (x) and the amount of eroded soil (y) is 0.65 then give a. The equation of the least squares regression line of yon x. b. What is the predicted amount of eroded soil for a flow rate of 10 L/s? 9. The correlation between amount of time spent studying for a final and the final exam grade is 0.9. Explain what this means.

Answer 1

7.

We have to perform Chi-square test for independence of two random variables owning a pet and survival.

(a)

We have to test for null hypothesis $\tiny H_0:\text{Owning a pet and survival are independent}$

against the alternative hypothesis

H1 : Owning a pet and survival are not independent

(b)

Here,

Number of rows $\tiny m=2$

Number of columns $\tiny n=2$

Degrees of freedom

v= m - 1)(n-1) = 1

$\tiny \therefore \text{p-value}=P\left(\chi^2_{1}>8.85\right)=0.002930889$ [Using R-code '1-pchisq(8.85,1)']

We reject our null hypothesis if $\tiny \text{p-value}<\alpha$ , level of significance.

We generally test for level of significance 0.10, 0.05, 0.01 or something like these.

So, we reject our null hypothesis.

Hence, based on the given data we can conclude that there is significant evidence that owning a pet and survival are not independent.

8.

(a)

Regression equation of y on x is given by

$\tiny y-\bar y=b_{yx}\left(x-\bar x\right)$

We have,

• $\tiny \bar x=6.1$
• $\tiny \bar y=4.2$
• $\tiny Var\left ( X \right )=0.5^2=0.25$
• $\tiny Var\left ( Y \right )=1.2^2=1.44$
• $\tiny r_{x,y}=0.65$

We know,

$\tiny r_{x,y}=\frac{Cov(X,Y)}{\sqrt {Var(X)*Var(Y)}}$

$\tiny \Rightarrow Cov(X,Y)=r_{x,y}\sqrt {Var(X)*Var(Y)}$

$\tiny \therefore Cov(X,Y)=0.65*0.5*1.2=0.39$

$\tiny \therefore b_{yx}=\frac{Cov(X,Y)}{Var(X)}=\frac{0.39}{0.25}=1.56$

$\tiny \therefore y-4.2=1.56\left(x-6.1\right)$

$\tiny \Rightarrow y-4.2=1.56x-1.56*6.1$

$\tiny \Rightarrow y=1.56x+4.2-9.516$

$\tiny \therefore y=1.56x-5.316$

(b)

For $\tiny x=10$ , predicted amount of eroded soil is given by

$\tiny \hat y=1.56*10-5.316=10.284$

9.

Our correlation coefficient is numerically large enough (almost close to highest possible value 1) as well as it posses positive value. It suggests that there is a strong positive linear correlation between time spent for a final examination and final examination grade. On average, with increase of time spent for a final examination, grade increases as well as with decrease of time spent for a final examination, grade decreases.