Question1. [Maximum mark: 6]
At a café, the waiting time between ordering and receiving a cup of coffee is dependent upon
the number of customers who have already ordered their coffee and are waiting to receive it.
Sarah, a regular customer, visited the café on five consecutive days. The following table
shows the number of customers, x , ahead of Sarah who have already ordered and are
waiting to receive their coffee and Sarah’s waiting time, y minutes.
Number of customers (x) | 3 | 9 | 11 | 10 | 5 |
| Sarah’s waiting time (y) | 6 | 10 | 12 | 11 | 6 |
The relationship between x and y can be modelled by the regression line of y on x with
equation y = ax + b.
(a) (i) Find the value of a and the value of b .
(ii) Write down the value of Pearson’s product-moment correlation coefficient, r. [3]
(b) Interpret, in context, the value of a found in part (a)(i). [1]
On another day, Sarah visits the café to order a coffee. Seven customers have already
ordered their coffee and are waiting to receive it.
(c) Use the result from part (a)(i) to estimate Sarah’s waiting time to receive her coffee.
▶️Answer/Explanation
1. (a) (i) a = 0.805084… and b = 2.88135…
a = 0.805 and b = 2.88
(ii)
r = 0.97777… r = 0.978
(b) a represents the (average) increase in waiting time (0.805 mins) per additional customer (waiting to receive their coffee) R1
(c) attempt to substitute x = 7 into their equation 8.51693..
8.52(mins)
Question
At a café, the waiting time between ordering and receiving a cup of coffee is dependent upon the number of customers who have already ordered their coffee and are waiting to receive it. Sarah, a regular customer, visited the café on five consecutive days. The following table shows the number of customers, x , ahead of Sarah who have already ordered and are waiting to receive their coffee and Sarah’s waiting time, y minutes.
The relationship between x and y can be modelled by the regression line of y on x with equation y = ax + b .
(a) (i) Find the value of a and the value of b
(ii) Write down the value of Pearson’s product-moment correlation coefficient, r . [3]
(b) Interpret, in context, the value of a found in part (a)(i). [1]
On another day, Sarah visits the café to order a coffee. Seven customers have already ordered their coffee and are waiting to receive it.
(c) Use the result from part (a)(i) to estimate Sarah’s waiting time to receive her coffee. [2]
▶️Answer/Explanation
Ans:
(a)(i) From graphing calculator, we have $y=0.805x+2.88$, i.e., $a=0.805$ and $b=2.88$.
(a)(ii) Again, from graphing calculator, $r=0.978$.
(b) For each increase in customer ($x$), the corresponding waiting time ($y$) increases by $a=0.805$ minutes.
(c) When $x=7$, we have $y=8.52$, thus, Sarah has to wait for $8.52$ minutes to receive her coffee.