In a chemical reaction the mass, $x$ grams, of a particular product at time $t$ minutes is given in this table.

The value of the product moment correlation coefficient is $0.9803$ correct to 4 significant figures. The scatter diagram for the data is shown below.

(a) Toby attempts to model the relationship between $x$ and $t$ with a straight line. Explain whether this is likely to provide a good model.

[1]

Toby now tries a model in which $x$ has been transformed to $\ln x$.

(b) (i) Sketch a scatter diagram of $\ln x$ against $t$ for the data given in the table.

[1]

(ii) Toby models the data with the equation $\ln x=c+d t$. Find the values of the constants $c$ and $d$ and state the value of the product moment correlation coefficient for this model.

[3]

(c) Comment on Toby’s two models.

[2]

b(ii) $d \approx 0.0573, c \approx 2.29, r \approx 0.999$

[Maximum marks: 7 marks]

(a)

The given product moment correlation coefficient suggests a strong positive linear correlation between the bivariate data. The given scatter diagram also suggest that as $t$ increases, $x$ increases increasingly and at the start, $x$ was increasing slower. Thus a linear model will not provide a good model.

(b)

(b)(ii)

Using GC, $d \approx 0.0573, c \approx 2.29, r \approx 0.999$

(c)

Both models suggest that there is a strong positive linear correlation between the bivariate data. However, the product moment correlation coefficient for model in (b) is closer to 1 and suggests a stronger linear correlation as compared to that for model in (a). Based on the given scatter diagram, as time passes, the mass should increase increasingly and thus model (B) is more appropriate.