Free preview 2/15

Question 1
A nutritionist records the daily sugar intake $S$ (in grams) and the corresponding energy level rating $E$ (on a scale of 1–10) for seven participants. The data are shown below.

Sugar intake $S$ (g) Energy rating $E$
10 8
20 7
30 6
40 5
50 4
60 3
70 2

Which of the following best describes the correlation between $S$ and $E$ ?
No clue? Show me the answer
Correct answerCorrect!Incorrect
CStrong negative linear correlation
Step-by-step walkthrough
Choose a solution method
Method #1Approach 1
Step 1: Identify the trend in the data
As sugar intake $S$ increases from 10 g to 70 g, the energy rating $E$ decreases consistently from 8 to 2. This indicates a negative direction.
Step 2: Assess the strength of the correlation
The data points decrease very consistently — every 10 g increase in $S$ corresponds to exactly a 1-unit drop in $E$ . The points would lie almost perfectly on a straight line, indicating strong correlation.
Step 3: Identify the form
The relationship appears linear since the decrease is uniform. The value of $r$ would be very close to $- 1$ .
Step 4: Select the correct description
Combining direction (negative), strength (strong), and form (linear), the correct description is strong negative linear correlation.
Method #2Approach 2
Step 1: Identify what is being asked
The question asks for the best description of the correlation between sugar intake and energy rating, considering direction, strength, and form.
Step 2: Eliminate 'Strong positive linear correlation'
As $S$ increases, $E$ decreases — not increases. A positive correlation would require both variables to increase together, so this option is incorrect.
Step 3: Eliminate 'Weak negative linear correlation'
Although the direction is correct (negative), the data is extremely consistent with very little scatter. This rules out 'weak' — the correlation is clearly strong.
Step 4: Eliminate 'No linear correlation'
There is an obvious and consistent downward linear trend in the data, so 'no linear correlation' is clearly incorrect.
Step 5: Select the correct answer
The remaining option, strong negative linear correlation, correctly captures the consistent downward trend in the data.
Question 2
A marine biologist measures the water depth $d$ (in metres) and the water temperature $T$ (in °C) at eight locations in a lake. The regression line of $T$ on $d$ is found to be $\hat{T} = - 0.85 d + 22.4$ . What is the predicted temperature at a depth of 10 metres?
No clue? Show me the answer
Correct answerCorrect!Incorrect
A $13.9$ °C
Step-by-step walkthrough
Choose a solution method
Method #1Approach 1
Step 1: Identify the regression equation and required substitution
The regression line is $\hat{T} = - 0.85 d + 22.4$ , and we need to predict $T$ when $d = 10$ .
Step 2: Substitute $d = 10$ into the equation
$\hat{T} = - 0.85 (10) + 22.4 = - 8.5 + 22.4 = 13.9$
Step 3: State the answer
The predicted temperature at a depth of 10 metres is 13.9 °C.
Method #2Approach 2
Step 1: Identify the calculation required
We must substitute $d = 10$ into $\hat{T} = - 0.85 d + 22.4$ and match the result to one of the four options.
Step 2: Eliminate $21.6$ °C
This would result from barely changing the intercept (e.g., subtracting only $0.8$ ), suggesting an arithmetic error. $- 0.85 \times 10 = - 8.5$ , not $- 0.8$ .
Step 3: Eliminate $14.4$ °C
$14.4 = - 8 + 22.4$ , which would result from using $- 0.8 \times 10$ instead of $- 0.85 \times 10$ . This is an incorrect coefficient.
Step 4: Eliminate $13.5$ °C
$13.5 = - 8.9 + 22.4$ , which doesn't arise from correct arithmetic with $- 0.85 \times 10 = - 8.5$ .
Step 5: Select the correct answer
$- 0.85 (10) + 22.4 = - 8.5 + 22.4 = 13.9$ °C, confirming $13.9$ °C is correct.

Free preview

13 more questions in this topic

Sugar intake

S

(g)

Energy rating

E

Step-by-step walkthrough

Step 1: Identify the trend in the data

Step 2: Assess the strength of the correlation

Step 3: Identify the form

Step 4: Select the correct description

Step 1: Identify what is being asked

Step 2: Eliminate 'Strong positive linear correlation'

Step 3: Eliminate 'Weak negative linear correlation'

Step 4: Eliminate 'No linear correlation'

Step 5: Select the correct answer

Step-by-step walkthrough

Step 1: Identify the regression equation and required substitution

Step 2: Substitute $d = 10$ into the equation

Step 3: State the answer

Step 1: Identify the calculation required

Step 2: Eliminate $21.6$ °C

Step 3: Eliminate $14.4$ °C

Step 4: Eliminate $13.5$ °C

Step 5: Select the correct answer

13 more questions in this topic

Step-by-step walkthrough

Step 1: Identify the trend in the data

Step 2: Assess the strength of the correlation

Step 3: Identify the form

Step 4: Select the correct description

Step 1: Identify what is being asked

Step 2: Eliminate 'Strong positive linear correlation'

Step 3: Eliminate 'Weak negative linear correlation'

Step 4: Eliminate 'No linear correlation'

Step 5: Select the correct answer

Step-by-step walkthrough

Step 1: Identify the regression equation and required substitution

Step 2: Substitute $d = 10$ into the equation

Step 3: State the answer

Step 1: Identify the calculation required

Step 2: Eliminate $21.6$ °C

Step 3: Eliminate $14.4$ °C

Step 4: Eliminate $13.5$ °C

Step 5: Select the correct answer

13 more questions in this topic