Question

A researcher wants to know if the gestation period of an animal can be used to predict life expectancy. She collects the following data:

- Draw a scatter plot

- Use the formulas to compute the linear correlation coefficient between gestation period and life expectancy.

- Use the formulas to find the least-squares regression line

- Would it be reasonable to use the equation to predict the life expectancy if the gestation period are 10, 50, or 200 days, respectively gestation period are 10, 50, and 200 days

- The observed life expectancy of the animal who has gestation period of 63 days. Is this life expectancy above average or below average among all animals with gestation period of 63 days?

- Find the coefficient of determination R^2= , and interpret the statistical meaning of R^2

- Test whether a linear relation exists between the gestation period and life expectancy, at alpha= 0.05

H0: vs H1:

Check the assumptions of the test. Does the residual plot confirm that the relation between the gestation period and life expectancy is linear? Why?

Test statistic: , df=

P-value =

Conclusion:

- If the null hypothesis is rejected, find a 95% confidence interval of beta1:

Animal	Gestation Period	Life Expectancy
cat	63	11
chicken	22	7.5
dog	63	11
duck	28	10
goat	151	12
lion	108	10
parakeet	18	8
pig	115	10
rabbit	31	7
squirrel	44	9

Answer 1

Draw a scatter plot

܀ ܀ ܀ .' _20 _40 _60 _80 100120140_160 Gestation period

- Use the formulas to compute the linear correlation coefficient between gestation period and life expectancy.

Correlation coefficient (r ) =

Σ( – 1)(y – 9) Σα – )?Σ(y - y)2

r = 0.726

- Use the formulas to find the least-squares regression line

The dependent variable (y) is the life expectancy and the independent variable (x) is gestation period.

The regression line of Y on X

y = Bo + B12

Where $\beta_{1}$ is the slope = $r\frac{\sigma_{y}}{\sigma_{x}}$ =

$\beta_{0}$ is the intercept =
у — Box
=

Therefore the equation is

y = 7.87 + 0.026.

- Would it be reasonable to use the equation to predict the life expectancy if the gestation period are 10, 50, or 200 days, respectively gestation period are 10, 50, and 200 days.

It would be reasonable to predict life expectancy with gestation period of 50 days since they fall with the range of x. Whereas we would have to extrapolate for 10 and 200 days which might not provide accurate results.

- The observed life expectancy of the animal who has gestation period of 63 days. Is this life expectancy above average or below average among all animals with gestation period of 63 days?

Sorry I am unable to provide solution for this question since it's not clear what is asked.

- Find the coefficient of determination R^2= interpret the statistical meaning of R^2

$R=r^{2}$ = 0.53

The coefficient of determination is 53%. It determines the variability of the data that is explained by the model. Usually > 75% is considered good fit, that is, whether model fits the data accurately or not.

- Test whether a linear relation exists between the gestation period and life expectancy, at alpha= 0.05

H0: $\beta_{1}$ = 0 . No relation exists between the gestation period and life expectancy
vs

H1: $\beta_{1}$$\neq 0$ A relation exists between the gestation period and life expectancy

Check the assumptions of the test. Does the residual plot confirm that the relation between the gestation period and life expectancy is linear? Why?

Residual plot

20 40 60 80 100 120 140 160

Since the residual plot looks non-linear and scattered we can say that there exists a linear relation between the variables.

Test statistic: = 2.983

Bi - Bi V Srr

Where $\hat{\beta_{1}}$ is the sample slope = 0.026

$\hat{\sigma^{2}}=(Syy-\frac{S^{2}xy}{Sxx}) \frac{1}{n-2}$

Where Syy = $(n-1)Sy^{2}$

Sxx = $(n-1)Sx^{2}$

Sxy = $\sum xy-n\bar{x}\bar{y}$

, df= n-2 = 8

P-value = P ($t_{n-2>T.S.}$)

P-value = 0.0175

Since p - value < 0.05

Conclusion: We reject the null hypo at 0.05 level of significance. There exists a relationship between the gestation period and life expectancy.

- If the null hypothesis is rejected, find a 95% confidence interval of beta1:

$(\hat{\beta_{1}}\bar{+}t_{n-2,\alpha/2}\sqrt{\frac{\sigma^{2}}{Sxx}})$

= 2.31

t8,0.05/2

$(0.026\bar{+}2.31*0.0087)$

CI = (0.0058, 0.0466)