A teacher took a random sample of 8 children from a class. For each child the teacher recorded the length of their left foot, f cm, and their height, h cm. The results are given in the table below. | f | 23 | 26 | 23 | 22 | 24 | 20 | 21 | | h | 135 | 134 | 136 | 140 | 134 | 130 | 132 | (You may use $\sum f = 186$, $\sum h = 1085$, $ S_{ff} = 39.5$, $S_{hh} = 139.875$, $\sum h = 25291$) (a) Calculate $ S_{fh} $ (b) Find the equation of the regression line of $ h $ on $ f $ in the form $ h = a + bf $. Give the value of $ a $ and the value of $ b $ correct to 3 significant figures. (c) Use your equation to estimate the height of a child with a left foot length of 25 cm. (d) Comment on the reliability of your estimate in (c), giving a reason for your answer. The left foot length of the teacher is 25 cm. (e) Give a reason why the equation in (b) should not be used to estimate the teacher's height.

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A teacher took a random sample of 8 children from a class - Edexcel - A-Level Maths Statistics - Question 7 - 2011 - Paper 2

Question 7

A teacher took a random sample of 8 children from a class. For each child the teacher recorded the length of their left foot, f cm, and their height, h cm. The resul... show full transcript

Worked Solution & Example Answer:A teacher took a random sample of 8 children from a class - Edexcel - A-Level Maths Statistics - Question 7 - 2011 - Paper 2

Step 1

Calculate $ S_{fh} $

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Answer

To find the value of $S_{fh}$ , we can use the formula:

$S_{fh} = \sum (f_i h_i) - \frac{(\sum f)(\sum h)}{n}$

Where:

$\sum f = 186$
$\sum h = 1085$
$n = 8$

Now, calculating:

$S_{fh} = (23 \times 135) + (26 \times 134) + (23 \times 136) + (22 \times 140) + (24 \times 134) + (20 \times 130) + (21 \times 132)$

Calculating this gives: $S_{fh} = 3105 + 3484 + 3128 + 3080 + 3216 + 2600 + 2772 = 18485$

Now substituting back in: $S_{fh} = 18485 - \frac{(186)(1085)}{8}$

Compute: $S_{fh} = 18485 - 20065.625 \approx -1580.625$

So, the value of $S_{fh}$ is approximately -1580.625.

Step 2

Find the equation of the regression line of $ h $ on $ f $ in the form $ h = a + bf $

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Answer

Using the formulas for the regression line:

Calculate slope $b$ : $b = \frac{S_{fh}}{S_{ff}}$

Given:

$S_{ff} = 39.5$
$S_{fh} \approx -1580.625$

So, $b = \frac{-1580.625}{39.5} \approx -40.00$

Calculate intercept $a$ : $a = \frac{\sum h}{n} - b \cdot \frac{\sum f}{n}$

Substituting the required values: $a = \frac{1085}{8} - (-40.00) \cdot \frac{186}{8}$ $a \approx 135.625 + 993 = 1128.625$

Thus, the regression equation is: $h = 1128.625 - 40.00 f$

Step 3

Use your equation to estimate the height of a child with a left foot length of 25 cm.

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Answer

Using the regression equation: $h = 1128.625 - 40.00 \times 25$

Compute this: $h \approx 1128.625 - 1000 = 128.625$

Thus, the estimated height of a child with a left foot length of 25 cm is approximately 128.63 cm.

Step 4

Comment on the reliability of your estimate in (c), giving a reason for your answer.

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Answer

The estimate is likely reliable because the left foot length of 25 cm falls within the range of the recorded foot lengths in the dataset. However, it's essential to note that this estimate has limitations. The sample size is small and may not represent the whole population, potentially skewing the results.

Step 5

Give a reason why the equation in (b) should not be used to estimate the teacher's height.

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Answer

The equation derived in (b) is based on the children’s data. It may not be appropriate to apply it to adults, as growth rates and body proportions can differ significantly between children and adults. Additionally, the teacher's height may not fit within the derived range, reducing the validity of any such estimates.

A teacher took a random sample of 8 children from a class - Edexcel - A-Level Maths Statistics - Question 7 - 2011 - Paper 2

Worked Solution & Example Answer:A teacher took a random sample of 8 children from a class - Edexcel - A-Level Maths Statistics - Question 7 - 2011 - Paper 2

Calculate $ S_{fh} $

Find the equation of the regression line of $ h $ on $ f $ in the form $ h = a + bf $

Use your equation to estimate the height of a child with a left foot length of 25 cm.

Comment on the reliability of your estimate in (c), giving a reason for your answer.

Give a reason why the equation in (b) should not be used to estimate the teacher's height.

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