Question

4) The relationship between the price (y) of an alloy ingot and the percentage of gold (x) contained was investigated. The data collected is shown below: % gold (x) 10 20 30 35 40 50 60 70 Price (y) 35 55 40 100 60 90 110 130 a) Plot a scatter diagram of the data. (3 marks) b) Explain what type of correlation is indicated by the scatter diagram. Give a reason for your answer. (2 marks) c) Calculate the product moment correlation coefficient (PMCC) for this data. (4 marks) d) Use the value of the PMCC to comment on: • the type and strength of correlation • the relationship between the % of gold and the price of the ingot. (3 marks) [Total 12 marks]

Answer 1

Answer a:
The Scatter – Diagram of the data is given below –

140 70, 130 120 60, 110 100 35, 100 50,90 80 Pricely) 60 40, 60 20,55 40 30, 40 10,35 20 0 0 LO 20 30 50 60 70 80 40 % gold(x)

Answer b:
The Scatter – Diagram indicates a positive correlation between the two variables.

It is because the scatter diagram is showing an increasing trend, that is, as the value of x increases the value of y also increases.

Answer c:
The following table shows all the calculations –

	% gold(x)	Price (y)	x^2	y^2	xy
	10	35	100	1225	350
	20	55	400	3025	1100
	30	40	900	1600	1200
	35	100	1225	10000	3500
	40	60	1600	3600	2400
	50	90	2500	8100	4500
	60	110	3600	12100	6600
	70	130	4900	16900	9100
Total	315	620	15225	56550	28750

Total number of observations, n = 8

Mean of x, $\bar{x}$ = 315/8 = 39.375

Mean of y, $\bar{y}$ = 620/8 = 77.5

Standard Deviation of x, Sx = {($\sum$(x^2)/n) - ($\bar{x}$^2)}^0.5 = {(15225/8) – (39.375^2)}^0.5 = 18.7812

Standard Deviation of y, Sy = {($\sum$(y^2)/n) - ($\bar{y}$^2)}^0.5 = {(56550/8) – (77.5^2)}^0.5 = 32.5960

Covariance between x and y, Cov.(x, y) = ($\sum$xy / n) - ($\bar{x}$$\bar{y}$) = (28750/8) – (39.375 x 77.5) = 542.1875

Product Moment Correlation Coefficient, r = Cov.(x, y) / (Sx.Sy) = 542.1875 / (18.7812 x 32.5960)

                                                                               = 0.8856

(All the measures are rounded up to 4 decimal places)

Answer d:
The Strength of the correlation is 0.8856, or close to 0.9, which indicates a strong relationship. Since, the value of the PMCC is positive, it indicates a positive relationship.

The % of gold and the price of the ingot are positively related because as the % of gold contained increases the price of the ingot also increases. The PMCC value also indicates the same thing that both are strongly related.