Question

A business manager wanted to know if there was a relationship between daily temperature measured in degrees Fahrenheit and number of employees who called-in sick each day. He collected the following data over 10 days:

Daily Temperature: 64 60 67 63 60 64 72 66 60 69

# Called-in Sick: 33 32 30 37 34 28 31 27 31 26

Use the Daily Temperature (X-variable) and Called-in Sick (Y-Variable) Data to answer the following questions.

a. What is the value of the slope of the regression line?

b. What is the value of the y-intercept?

c. Provide an interpretation of your answer to part B.

d. What is the final form of the regression line (i.e., y’ = a + b(x))?

e. Use Excel to obtain a scatterplot of the data with the regression line embedded in it.

f. What is the predicted number of employees that will call-in sick on a day that is 71 degrees Fahrenheit?

Answer 1

			$(x-\bar{x})(y-\bar{y})$	$(x-\bar{x})^2$
	64	33	-3.36	2.56
	60	32	-6.16	31.36
	67	30	-1.26	1.96
	63	37	-15.86	6.76
	60	34	-17.36	31.36
	64	28	4.64	2.56
	72	31	0.64	40.96
	77	27	-44.46	129.96
	60	31	-0.56	31.36
	69	26	-16.66	11.56
Total	656	309	-100.4	290.4
Total/n	65.6	30.9

See that the last column, $\bar x= 65.6, \bar y= 30.9$

Note that the least square estimates for a and b in Y= a+bx, are:

$\hat{b}=\frac{\sum_{i=1}^n (x_i-\bar x)(y_i-\bar y)}{\sum_{i=1}^n (x_i-\bar x)^2}= \frac{-100.4}{290.4}=-0.3457$

and $\hat a= \bar{y}-\hat b\bar{x}=53.5779$

a. What is the value of the slope of the regression line?

Ans. b= -0.3457

b. What is the value of the y-intercept?

Ans. a= 53.5779

c. Provide an interpretation of your answer to part b.

Ans. If X=0, that is, if the temperature would be 0 Fahrenheit, then the average number of employees who call in sick would be 53 approximately.

d. What is the final form of the regression line ?

Ans. Y= 53.5779 - 0.3457 x

e. Use Excel to obtain a scatterplot of the data with the regression line embedded in it.

Ans. Steps to obtain scatterplot with regression line embedded in it:

1) select columns of X and Y.

2) Go to insert -> charts -> scatter (X,Y)

3)Right click on data points, and select add trend line -> linear (you can also insert the trendline equation on the graph itself)

Scatterplot of Y on x y = -0.3457x53.58% Called-in-sick 20 80 100 40 60 Daily Temperature

f. What is the predicted number of employees that will call-in sick on a day that is 71 degrees Fahrenheit?

Ans. x=71,

then Y= 53.5779 - 0.3457 *71= 29 nearly.