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 Correlation & Regression: 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 $S_t$ , we calculate:

$S_t = \frac{\sum t^2 - \left(\frac{\sum t}{n}\right)^2}{n-1}$

Given that:

$\sum t = 2688, \quad n = 10$

We can find:

$S_t = \frac{2688^2 - \left(\frac{2688}{10}\right)^2}{10-1} = 192$

Similarly, for $S_w$ , we have:

$S_w = \frac{\sum w^2 - \left(\frac{\sum w}{n}\right)^2}{n-1}$

With:

$\sum w = 111.75$

The calculation yields:

$S_w = \frac{1760.62 - \left(\frac{111.75}{10}\right)^2}{9} = 5.03$

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 is given by:

$r = \frac{S_{tw}}{\sqrt{S_t S_w}}$

Substituting known values:

$r = \frac{0.16}{\sqrt{192 \times 5.03}} = -0.908469$

Thus, to three significant figures, $r \approx -0.908$ .

Step 3

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

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Answer

The regression line can be derived using:

$b = \frac{S_{tw}}{S_t}$

Calculating:

$b = \frac{0.16}{192} = -0.0263$

Then, using:

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

Where $ar{w} = \frac{111.75}{10} = 11.175$ and $ar{t} = \frac{2688}{10} = 268.8$ :

$a = 11.175 - (-0.0263 \times 268.8) \approx 11.59$

Thus, the equation is:

$w = 11.59 - 0.0263t$ .

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. This is because weight ( $w$ ) is dependent on age ( $t$ ), illustrating that as age varies, the weight is affected.

Step 5

Using this model, estimate the weight of a coin which is 5 years old

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Answer

Using the regression equation:

$w = 11.59 - 0.0263(5)$

We calculate:

$w \approx 11.59 - 0.1315 \approx 11.46$

Thus, the estimated weight of a 5-year-old coin is approximately 11.46 grams.

Step 6

Using this model, estimate the effect of an increase of 4 years in age on the weight of a coin

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Answer

For an increase of 4 years, we apply:

$w_{new} = w_{old} - b \cdot 4$

This leads to:

$w_{new} = 11.46 - (-0.0263)\cdot 4 \approx 11.46 + 0.1052 \approx 11.57$

Thus, the weight increases by approximately 0.1052 grams.

Step 7

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

Removing the fake coin, which is significantly different in weight at 20 grams, would likely increase the value of the product moment correlation coefficient. This is because the fake coin creates distortion in the data, and its removal would allow the correlation to more accurately reflect the relationship between age and weight.

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 the weight of a coin which is 5 years old

Using this model, estimate the effect of an increase of 4 years in age on the weight of a coin

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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