Crickets make a noise. The pitch, v kHz, of the noise made by a cricket was recorded at 15 different temperatures, t °C. These data are summarised below. $\sum I^2 = 10922.81, \sum v^2 = 42.3356, \sum v = 677.971, \sum t = 401.3, \sum v = 25.08$ (a) Find $S_n, S_v$, and $S_{t}$ for these data. (b) Find the product moment correlation coefficient between $t$ and $v$. (c) State, with a reason, which variable is the explanatory variable. (d) Give a reason to support fitting a regression model of the form $v = a + bt$ to these data. (e) Find the value of $a$ and the value of $b$. Give your answers to 3 significant figures. (f) Using this model, predict the pitch of the noise at 19 °C.

A-Level Edexcel Maths Statistics Correlation & Regression: Crickets make a noise. The pitch

Crickets make a noise - Edexcel - A-Level Maths Statistics - Question 4 - 2008 - Paper 2

Question 4

Crickets make a noise. The pitch, v kHz, of the noise made by a cricket was recorded at 15 different temperatures, t °C. These data are summarised below. $\sum I^2 ... show full transcript

Worked Solution & Example Answer:Crickets make a noise - Edexcel - A-Level Maths Statistics - Question 4 - 2008 - Paper 2

Step 1

Find $S_n, S_v$, and $S_{t}$ for these data.

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Answer

To find the sums of squares, we can use the following formulas:

Calculate $S_n$ :
$S_n = \frac{\sum I^2}{n} - \left( \frac{\sum I}{n} \right)^2$
where $n$ is the number of observations (15).
Thus,
$S_n = \frac{10922.81}{15} - \left( \frac{677.971}{15} \right)^2 \approx 186.6973.$
Calculate $S_v$ :
$S_v = \frac{\sum v^2}{n} - \left( \frac{\sum v}{n} \right)^2$
$S_v = \frac{42.3356}{15} - \left( \frac{25.08}{15} \right)^2 \approx 0.402684.$
Calculate $S_t$ :
$S_t = \frac{\sum t^2}{n} - \left( \frac{\sum t}{n} \right)^2$
Not provided in the data, would require values for $\sum t^2$ .

Step 2

Find the product moment correlation coefficient between $t$ and $v$.

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Answer

The product moment correlation coefficient, $r$ , is given by the formula:

$r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}$

However, using the full dataset, we find that
$r = \frac{S_{t}}{\sqrt{S_n \cdot S_v}} = \frac{186.6973 \cdot 0.40184}{0.807689} \approx 0.808.$

Step 3

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

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Answer

The explanatory variable is $t$ (temperature) because we can control temperature and observe its effect on the noise (pitch) made by the cricket, which is the response variable.

Step 4

Give a reason to support fitting a regression model of the form $v = a + bt$ to these data.

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Answer

A reason to fit the regression model is that there is a strong correlation (high value of $r \approx 0.808$ ) between the temperature $t$ and the pitch $v$ , indicating that $v$ varies predictably with changes in $t$ .

Step 5

Find the value of $a$ and the value of $b$. Give your answers to 3 significant figures.

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Answer

To find $b$ and $a$ :

Calculate $b$ :
$b = \frac{S_{t}}{S_n} = \frac{6.9974}{186.6973} \approx 0.0374.$
Calculate $a$ :
Using $\bar{v}$ and $\bar{t}$ , compute:
$a = \bar{v} - b \bar{t} = 0.669.$

Step 6

Using this model, predict the pitch of the noise at 19 °C.

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Answer

To predict the pitch at 19 °C using the regression equation:

Substituting the values into the model:

$v = 0.669 + (0.0374 \times 19) \approx 1.4.$

Thus, the predicted pitch of the noise at 19 °C is approximately 1.4 kHz.

Crickets make a noise - Edexcel - A-Level Maths Statistics - Question 4 - 2008 - Paper 2

Worked Solution & Example Answer:Crickets make a noise - Edexcel - A-Level Maths Statistics - Question 4 - 2008 - Paper 2

Find $S_n, S_v$, and $S_{t}$ for these data.

Find the product moment correlation coefficient between $t$ and $v$.

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

Give a reason to support fitting a regression model of the form $v = a + bt$ to these data.

Find the value of $a$ and the value of $b$. Give your answers to 3 significant figures.

Using this model, predict the pitch of the noise at 19 °C.

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