The number of reported cases, $x$ (in hundreds) of a virus in the ${{n}^{\text{th}}}$ month are

Question

The number of reported cases, $x$ (in hundreds) of a virus in the ${{n}^{\text{th}}}$ month are given in the table below. $n$ $1$ $2$ $3$ $4$ $5$ $6$ $x$ $0.26$ $3.05$ $4.12$ $4.63$ $5.15$ $k$ It is given that the least squares regression line of $x$ on $n$ is $x=0.943n+0.484$. Show that the value of $k$ to two decimal places is $5.50$. [1] Use the value of $k$ in part (i) for the rest of the question.[1] Draw a scatter diagram to illustrate the data. [1] Explain why a linear model is inappropriate. [1] Explain clearly which of the following models (A) or (B), where $a$ and $b$ are positive constants, is more appropriate to model these values. (A) $x=a+b\ln n$, (B) $x=a+b{{n}^{2}}$. Use the model that you have identified, calculate the values of $a$ and $b$, and use them to predict the month in which the number of reported cases is at least $1000$. [1] Give an interpretation, in context, of the value of $a$ for the model you have identified in part (iv).[1]

Accepted Answer

(iii) From the scatter plot, $x$ and $n$ have a curvilinear relationship. Therefore, a linear model is inappropriate even though the product moment correlation coefficient is relatively high (i.e. $0.915$). (iv) The graph of $x=a+b{{n}^{2}}$ is concave upwards (or increase at increasing rate). The graph of $x=a+b\ln n$ is concave downwards (or increase at decreasing rate) similar to the scatter plot. From GC, ${{r}_{B}}=0.8212061688\approx 0.821$ ($3$ s.f.);${{r}_{A}}=0.9859197289\approx 0.986$ ($3$ s.f.) which is closer to $1$ and hence suggested a relatively stronger linear relationship between $x$ and $\ln n$ as compared to $x$ and ${{n}^{2}}$. Therefore Model (A) is more appropriate; $27$ (v) The value of $a$ represents the estimated number of cases (in hundreds) of the virus reported in the first month.

2015 HCI P2 Q9

2015 HCI P2 Q9