Question

The cost of a leading liquid laundry detergent in different sizes is given below. Size (ounces) 16 32 64 200 Cost ($) 3.49 4.39 5.29 10.29 + Part (a) Part (b) Part (c) Calculate the least squares line. Put the equation in the form of ý = a + bx. (Round your answers to three decimal places.) ŷ = Part (d) Find the correlation coefficient r. (Round your answer to four decimal places.) Is it significant? Yes O No Part (0) If the laundry detergent were sold in a 50-ounce size, find the estimated cost. (Use your equation from part (c). Round your answer to two decimal places.) Part (1) If the laundry detergent were sold in an 80-ounce size, find the estimated cost. (Use your equation from part (c). Round your answer to two decimal places.) $ Part (9) Part (h) + Part (0) Part (1) What is the slope of the least squares (best-fit) line? (Round your answer to three decimal places.) Interpret the slope. (Round your answer to three decimal places.) As the number of ounces of liquid laundry detergent increases by one unit, the detergent cost increases by dollars

Answer 1

Part (C)

Computational Table:

	Size (X)	Cost (Y)	X2	Y2	XY
	16	3.49	256	12.18	55.84
	32	4.39	1024	19.27	140.48
	64	5.29	4096	27.98	338.56
	200	10.29	40000	105.88	2058
Total	312	23.46	45376	165.3204	2592.88

Calculation:

312 Χ. ΣΧί = 78 η 4

Υ ΣΥ 23.46 4 5.865 n

For Slope:

ο = ηΣΧΥ (ΣΧ)(ΣΥ) ηΣΧ2 – (ΣΧ)2

b 4 * 2592.88 – (312) (23.46) 4 * 45376 – (312)

b = 0.036

For Intercept:

q=Y - 6.X

a = 5.865 - 0.036 * 78

a = 3.036

Therefore, the least square regression line would be,

Ý = a+b(X)

$\hat{Y}$ = 3.036 + 0.036 (X)

d)

Correlation coefficient (r):

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

$r = \frac{4*2592.88 - (312)(23.46)}{\sqrt{4*45376 - (312)^2}\sqrt{4*165.32 - (23.46)^2}}$

r = 0.9990

Is it significant,

Correlation coefficient (r) value is near to +1, i.e. there is Strong Postive Correlation between size (Ounces) and Cost ($)

ANSWER: A
A. Yes

Part (e)

The least square regression line would be,

Ý = a+b(X)

$\hat{Y}$ = 3.036 + 0.036 (X)

Put X = 50

$\hat{Y}$ = 3.036 + 0.036*50

$\hat{Y}$ = 4.85

Part (f)

The least square regression line would be,

Ý = a+b(X)

$\hat{Y}$ = 3.036 + 0.036 (X)

Put X = 80

$\hat{Y}$ = 3.036 + 0.036*80

$\hat{Y}$ = 5.94

i)

For Slope:

ο = ηΣΧΥ (ΣΧ)(ΣΥ) ηΣΧ2 – (ΣΧ)2

b 4 * 2592.88 – (312) (23.46) 4 * 45376 – (312)

b = 0.036

As the number of ounces of liquid laundry detergent increased by one unit, the detergent cost increased by 0.036 dollars.