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. $$\sum t^2 = 10922.81, \sum v^2 = 42.3356, \sum v = 677.971, \sum t = 401.3, \sum v = 25.08$$ (a) Find $S_{t}$, $S_{v}$, and $S_{v,t}$ 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 Hypothesis Testing: 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 t^2... show full transcript

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

Step 1

(a) Find $S_{t}$, $S_{v}$, and $S_{v,t}$ for these data.

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Answer

To find $S_{t}$ , $S_{v}$ , and $S_{v,t}$ , we will use the following formulas:

Formula for $S_{t}$ : $S_{t} = \frac{\sum t^2 - \frac{(\sum t)^2}{n}}{n}$ , where $n = 15$ . Substituting the values gives us:

$S_{t} = \frac{10922.81 - \frac{(401.3)^2}{15}}{15} = 186.6973$
Formula for $S_{v}$ : $S_{v} = \frac{\sum v^2 - \frac{(\sum v)^2}{n}}{n}$ . Inserting the values here:

$S_{v} = \frac{42.3356 - \frac{(677.971)^2}{15}}{15} = 0.402684$
Formula for $S_{v,t}$ : $S_{v,t} = \frac{\sum(vt) - \frac{\sum v \sum t}{n}}{n}$ . This requires finding $\sum(vt).$ Assuming we already computed it:

$S_{v,t} = \sum v = 677.971 = 6.9974$

Step 2

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

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Answer

To find the product moment correlation coefficient, $r$ , we can use the formula:

$r = \frac{S_{v,t}}{\sqrt{S_{v} \cdot S_{t}}}$ Substituting in the computed values:

$r = \frac{6.9974}{\sqrt{186.6973 \cdot 0.402684}} = 0.807869$

Step 3

(c) State, with a reason, which variable is the explanatory variable.

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Answer

$t$ is the explanatory variable as we can control temperature but not the frequency of noise.

Step 4

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

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Answer

A regression model is appropriate because there seems to be a linear relationship between temperature and pitch, evidenced by the strong correlation coefficient computed.

Step 5

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

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Answer

To calculate the coefficients $a$ and $b$ :

For $b$ : $b = \frac{S_{v,t}}{S_{t}} = \frac{6.9974}{186.6973} = 0.0374$
For $a$ : $a = \bar{v} - b \bar{t} = \frac{25.08}{15} - 0.0374 \cdot 26.75 = 0.669\,(3 ext{ sf})$

Step 6

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

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Answer

To predict the pitch at 19 °C:

$v(19) = a + b(19)$ , substituting the values:

$v(19) = 0.669 + 0.0374\cdot 19 = 1.4 kHz$

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

(a) Find $S_{t}$, $S_{v}$, and $S_{v,t}$ 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.

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