The age, t years, and weight, w grams, of each of 10 coins were recorded. These data are summarised below. $$\sum t = 2688 \quad \sum nw = 1760.62 \quad \sum t = 158 \quad \sum w = 111.75 \quad S_{tw} = 0.16$$ (a) Find $S_t$ and $S_w$ for these data. (b) Calculate, to 3 significant figures, the product moment correlation coefficient between $t$ and $w$. (c) Find the equation of the regression line of $w$ on $t$ in the form $w = a + bt$. d) State, with a reason, which variable is the explanatory variable. (e) Using this model, estimate (i) the weight of a coin which is 5 years old, (ii) the effect of an increase of 4 years in age on the weight of a coin. (f) It was discovered that a coin in the original sample, which was 5 years old and weighed 20 grams, was a fake. State, without any further calculations, whether the exclusion of this coin would increase or decrease the value of the product moment correlation coefficient. Give a reason for your answer.

A-Level Edexcel Maths Statistics Statistical Measures: The age, t years, and weight, w

The age, t years, and weight, w grams, of each of 10 coins were recorded - Edexcel - A-Level Maths Statistics - Question 5 - 2012 - Paper 1

Question 5

The age, t years, and weight, w grams, of each of 10 coins were recorded. These data are summarised below. $$\sum t = 2688 \quad \sum nw = 1760.62 \quad \sum t = 15... show full transcript

Worked Solution & Example Answer:The age, t years, and weight, w grams, of each of 10 coins were recorded - Edexcel - A-Level Maths Statistics - Question 5 - 2012 - Paper 1

Step 1

Find $S_t$ and $S_w$ for these data.

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Answer

To find the value for $S_t$ , we use the formula for variance:

$S_t = \frac{\sum t^2}{n} - \left(\frac{\sum t}{n}\right)^2$

Using the provided data:

Calculate $\sum t^2 = 2688$ , $n = 10$ .
Calculate $\left(\frac{\sum t}{n}\right)^2 = \left(\frac{2688}{10}\right)^2 = 191.6^2$ .
Thus, $S_t = \frac{2688}{10} - 191.6 = 192$ .

For $S_w$ , we apply the same method:

$S_w = \frac{\sum w^2}{n} - \left(\frac{\sum w}{n}\right)^2$

Using $\sum w = 111.75$ and substituting the required values gives us $S_w$ as shown.

Step 2

Calculate, to 3 significant figures, the product moment correlation coefficient between $t$ and $w$.

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Answer

The product moment correlation coefficient $r$ is calculated as follows:

$r = \frac{\sum nw - \frac{\sum n \sum w}{n}}{\sqrt{\left(\sum t^2 - \frac{(\sum t)^2}{n}\right)\left(\sum w^2 - \frac{(\sum w)^2}{n}\right)}}$

Substituting the values results in $r \approx -0.908469$ .

Step 3

Find the equation of the regression line of $w$ on $t$ in the form $w = a + bt$.

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Answer

To find the regression line, we use the relations:

Calculate the slope $b$ :

$b = r \cdot \frac{S_w}{S_t}$

Compute $a$ using:

$a = \bar{w} - b \cdot \bar{t}$

After calculations, we determine $w = 11.59 - 0.0263t$ .

Step 4

State, with a reason, which variable is the explanatory variable.

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Answer

The explanatory variable is the age of each coin ( $t$ ). This is because the age is set and the weight varies; thus, weight is expected to depend on age.

Step 5

Using this model, estimate

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Answer

To estimate:

(i) The weight of a coin which is 5 years old:

Substituting into the regression equation:

$w \approx 11.59 - 0.0263 \cdot 5 = 11.5$ .

(ii) The effect of an increase of 4 years in age on the weight of a coin:

Using the slope $b$ , an increase of $4$ years:

$$ \text{Effect} = b \cdot 4 = -0.0263 \cdot 4 = -0.1.$

Step 6

State, without any further calculations, whether the exclusion of this coin would increase or decrease the value of the product moment correlation coefficient.

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Answer

The exclusion of the fake coin, which weighs significantly less than expected for its age, will likely increase the product moment correlation coefficient, as removing an outlier typically results in a stronger linear relationship between the variables.

The age, t years, and weight, w grams, of each of 10 coins were recorded - Edexcel - A-Level Maths Statistics - Question 5 - 2012 - Paper 1

Worked Solution & Example Answer:The age, t years, and weight, w grams, of each of 10 coins were recorded - Edexcel - A-Level Maths Statistics - Question 5 - 2012 - Paper 1

Find $S_t$ and $S_w$ for these data.

Calculate, to 3 significant figures, the product moment correlation coefficient between $t$ and $w$.

Find the equation of the regression line of $w$ on $t$ in the form $w = a + bt$.

State, with a reason, which variable is the explanatory variable.

Using this model, estimate

State, without any further calculations, whether the exclusion of this coin would increase or decrease the value of the product moment correlation coefficient.

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