Mathematics 9709/61 — May/June 2017

Walkthrough

We are told that the sum of the coded values $x-k$ is 315. Since there are 30 buns, the mean of these coded values is

$$\bar{y} = \frac{315}{30} = 10.5$$

Each original weight is $k$ more than its coded value, so the original mean is the coded mean plus $k$:

$$\bar{x} = \bar{y} + k$$

The original mean is given as 50.5, so

$$50.5 = 10.5 + k$$

and therefore $k = 40$. Another way to see this is to multiply the mean by 30 to get $\sum x = 1515$, then solve $1515 - 30k = 315$.

Key Takeaways

Coding data by subtracting a constant shifts the mean by that same constant. The mean of coded data can be used to find the original mean, or to find the coding constant when the original mean is known.

Common Mistakes

• Dividing 315 by 30 is essential; using 315 directly as the coded mean is wrong.
• After finding $1515 - 315 = 1200$, some students give 1200 as the answer, but this is $30k$, not $k$; it must be divided by 30.
• A sign error in $\sum(x-k) = \sum x - 30k$ can lead to $k = 61$ instead of 40.

Things to Be Careful About

• Use $n = 30$ as the denominator when finding the coded mean.
• Remember that subtracting $k$ from every value lowers the mean by $k$, so $\bar{x} = \bar{y} + k$.
• Check the answer: $50.5 - 10.5 = 40$ confirms the value of $k$.

Walkthrough

Since $x = (x-k)+k$, each original weight is obtained from its coded value by adding the same constant $k$. Adding a constant shifts all values but does not change how spread out they are, so the standard deviation of $x$ is the same as the standard deviation of $x-k$. For the coded values $y = x-k$, the variance is

$$\text{Var}(y) = \frac{\sum y^2}{n} - \bar{y}^2$$

Here $\sum y^2 = 4022$, $n = 30$, and $\bar{y} = 10.5$. Therefore

$$\text{Var}(y) = \frac{4022}{30} - 10.5^2 = 134.0667 - 110.25 = 23.8167$$

Taking the square root gives

$$\text{sd} = \sqrt{23.8167} = 4.88 \text{ g}$$

Alternatively, expand $\sum(x-k)^2 = \sum x^2 - 2k\sum x + 30k^2$. With $k = 40$ and $\sum x = 1515$,

$$4022 = \sum x^2 - 80(1515) + 30(1600)$$ $$\sum x^2 = 77222$$

Then $\text{Var}(x) = \frac{77222}{30} - 50.5^2 = 23.8167$, giving the same standard deviation.

Key Takeaways

Variance measures spread using squared deviations, and the formula $\text{Var} = \frac{\sum y^2}{n} - \bar{y}^2$ can be applied to coded data. Since subtracting a constant does not affect spread, the standard deviation of coded data equals the standard deviation of the original data.

Common Mistakes

• Using $4022/30$ as the variance without subtracting $\bar{y}^2$.
• Subtracting $50.5^2$ when using the coded formula; the coded mean is $10.5$, not 50.5.
• Forgetting to take the square root at the end.
• Mixing coded sums with uncoded means in the same formula.

Things to Be Careful About

• Use $n = 30$ in the denominator, not 29.
• The variance is in grams squared; the standard deviation is in grams.
• Round only at the end to avoid rounding errors; the required value is 4.88 g.
• When expanding the uncoded sum of squares, keep the signs correct: $\sum(x-k)^2 = \sum x^2 - 2k\sum x + 30k^2$.