Question

1. A store manager wishes to find out whether there is a relationship between the age of her employees and the number of sick days they take each year. The data for the sample are shown. Age ()18 26 39 48 53 58 Days () 16 12 9 56 2 Draw the scatterplot and describe the association between the two variables. a) b) Is there a significant (nonzero) linear relationship between the age of the employees and the number of sick days? Use 5% significance level. [ldentify the null hypothesis, altemative hypothesis, test statistic, critical value from Table A-6, conclusion about the null hypothesis, and final conclusion. -45- MATH 160 Last Name First Name: Student ID # c) If there is a significant linear relationship, find the regression line. d) Find the best predicted number of sick days for an employee at 45 year of age.

Answer 1

Question 1

Here to calculate regression coefficient or the regression line coefficient, we require the value of $\sum x$ , $\sum y$ , $\sum xy$ , $\sum x^2$ , $\sum y^2$

	Age(x)	Days(y)	x^2	y^2	xy
	18	16	324	256	288
	26	12	676	144	312
	39	9	1521	81	351
	48	5	2304	25	240
	53	6	2809	36	318
	58	2	3364	4	116
Sum	242	50	10998	546	1625

Here as we can see the direction of scatter plot is negative and relationship seems strong.

(b) Here regression coefficient

r = [nΣxy - ΣxΣy]/ sqrt [n(Σx² - (Σx)²] [n(Σy² - (Σy)²]

r = [6 * 1625 - 242 * 50]/ sqrt [(6 * 10998 - 242²) * (6 * 546 - 50²)]

r = -2350/ sqrt [7424 * 776]

r = -2350/2400.31 = -0.9791

Here for n = 6

dF = n -2 = 6 - 2 = 4

t = r sqrt [(n-2) / (1 -r² )] = -0.9791 * sqrt [(6 - 2)/ (1 - 0.9791²)] = -9.6283

Here critical value for dF = 4

t_critical = 2.7764

Here t > t_critical  so we will reject the null hypothesis and conclude that there is significant relationship between the variables.

(c)

y^ = a + bx

a = [(Σy) (Σx² ) - (Σx) (Σxy)]/ [ n (Σx² ) - (Σx)² ]

a = [50 * 10998 - 242 * 1625]/ [6 * 10998 - 242²]

a = 21.1005

b =  [ n(Σxy) - (Σx)((Σy)]/ [ n (Σx² ) - (Σx)² ]

b = [6 * 1625 - 242 * 50]/ [6 * 10998 - 242²]

b= -2350/7424 = -0.3165

y^ = 21.1005 - 0.3165 x

(d) Here x = 45

y^ = 21.1005 - 0.3165 * 45 = 6.86 days