Answer to Question #265751 in Statistics and Probability for boo

Question

Consider the following data reflecting lengths of stay in the hospital (recorded in days) and the

total charge (in R1000) for six patients undergoing a minor surgical procedure:

Length of Stay 5 7 9 10 12 15

Total charge 6 5 7.2 8 9.4 7.9

Consider length of stay as the independent and total charge as the dependent variable.

(a) Describe the relationship between length of stay and total charge using a scatter diagram.

(b) Compute the sample correlation coefficient. (2 decimals)

(c) Compute the regression equation. (3 decimals)

(d) Estimate the total charge for an individual who stays 11 days in the hospital. (3 decimals)

(e) Suppose we compare two patients and one stays 3 days longer in the hospital than the other.

What is the expected difference in total charges between these patients?

Suppose the sample correlation coefficient is 𝑟 = 0.88. Is there evidence of a significant

correlation between the length of stay and the total charge? Run the appropriate test at 𝛼 = 0.05.

Answer 1

a)

total charge decreases for length of stay from 5 to 7

total charge increases for length of stay from 7 to 12

total charge decreases for length of stay from 12 to 15

b)

correlation coefficient:

$r=\frac{\sum(x_i-\overline{x})(y_i-\overline{y})}{\sqrt{\sum(x_i-\overline{x})^2(y_i-\overline{y})^2}}=0.74$

c)

regression equation:

$y=ax+b$

where y is total charge,

x is length of stay

$a=\frac{n\sum xy-\sum x\sum y}{n\sum x^2-(\sum x)^2}=0.325x$

$b=\frac{\sum y-a\sum x}{n}=4.106$

$y=0.325x+4.106$

d)

total charge for an individual who stays 11 days in the hospital:

$y(11)=0.325\cdot11+4.106=7.7$

e)

expected difference in total charges:

$y(x+3)-y(x)=0.325(x+3)+4.106-0.325x-4.106=0.325\cdot3=0.975$

$H_0:r=0$

$H_a:r\neq0$

$t=\frac{r\sqrt{n-2}}{1-r^2}=\frac{0.88\sqrt{6-2}}{1-0.88^2}=7.801$

$df=n-2=6-2=4$

critical value:

$t_{crit}=2.776$

Since $t>t_{crit}$ we reject the null hypothesis. There is evidence of a significant

correlation between the length of stay and the total charge.