Weight of a scoop in two ice cream parlours

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Stanislav Duris

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Chris Graham	said	Ready to use	7 years, 9 months ago
Stanislav Duris	said	Needs to be tested	7 years, 9 months ago
Elliott Fletcher	said	Has some problems	7 years, 9 months ago

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Chris Graham 7 years, 9 months ago

Gave some feedback: Ready to use

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Chris Graham commented 7 years, 9 months ago

Good question. I've made some minor changes to the advice.

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Stanislav Duris 7 years, 9 months ago

Published this.

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Stanislav Duris 7 years, 9 months ago

Saved a checkpoint:

As I said in one of our meetings, my main goal in creating this question was to show students that two equal means don't mean two equal distributions and to play around with the fact that all measures of average can be misleading when interpreted incorrectly. I started another question as advised by Chris, Christian and Lauren about salaries (many poor people vs few very rich residents leads to false beliefs about average salary in the area) but I don't think I'll manage to get it done up to a point where someone else can finish it, so I think someone else just needs to start that question from the scratch. A histogram could be included in the Salaries question. I tried to play around with Geogebra so I could include graphs in my future questions, but I did not have time to finish this successfully, so someone else may need to do this to cover the GCSE statistics.

Restore this checkpoint

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Stanislav Duris 7 years, 9 months ago

Gave some feedback: Needs to be tested

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Elliott Fletcher 7 years, 9 months ago

Gave some feedback: Has some problems

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Elliott Fletcher commented 7 years, 9 months ago

This is a very good question, there isn't really much i would change just a few minor things.

Advice

a) You don't need the "first" in the first sentence here as there is only one column for Sweet Heaven.

When you say "Similarly for Tasty Hell:" i think the colon should just be a comma but i'm not 100% sure.

b) You could maybe split up this paragraph by starting a new line after each sentence, but it's up to you.

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Stanislav Duris commented 7 years, 9 months ago

Thank you Chris, I have used [3pt].

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Stanislav Duris 7 years, 9 months ago

Gave some feedback: Needs to be tested

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Chris Graham commented 7 years, 9 months ago

Good question. Just a couple of things:

In the statement you say that the shops will be referred to as $s$ and $t$ , but this notation is not used in the question itself, just the advice, so why not introduce this shorthand at the start of the advice instead.

In your advice, you have several calculations involving fractions inside the align environment. This is fine, however you will notice that they appear quite squashed with the default line spacing. You could add [1pt] (play around with the number) immediately after the line-break (the \\) to add additional line spacing, e.g.

\overline{s} &= \frac{\sum s}{n} \\[2pt]
...

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Chris Graham 7 years, 9 months ago

Gave some feedback: Has some problems

○

Stanislav Duris 7 years, 9 months ago

Gave some feedback: Needs to be tested

○

Stanislav Duris 7 years, 9 months ago

Gave some feedback: Has some problems

○

Stanislav Duris 7 years, 9 months ago

Gave some feedback: Needs to be tested

○

Stanislav Duris 7 years, 9 months ago

Created this.

Name	Status	Author	Last Modified
Weight of a scoop in two ice cream parlours	Ready to use	Stanislav Duris	20/11/2019 14:45
Measures of central tendency-Q3	draft	Lauren Frances Desoysa	08/08/2018 13:08
Measures of central tendency from a list 2	Needs to be tested	Lauren Frances Desoysa	11/11/2020 10:03
Michael's copy of Weight of a scoop in two ice cream parlours	draft	Michael Proudman	07/12/2021 09:49

There are 2 other versions that do you not have access to.

Question statement

Two ice cream parlours called Sweet Heaven and Tasty Hell both sell ice cream for the same price. Alice likes both of these places equally, and has visited each place 10 times. After every visit, Alice measured the weight of her scoop, in grams, to the nearest integer. Here is the table of her values:

Sweet Heaven (g)	$\var{a[0]}$	$\var{a[1]}$	$\var{a[2]}$	$\var{a[3]}$	$\var{a[4]}$	$\var{a[5]}$	$\var{a[6]}$	$\var{a[7]}$	$\var{a[8]}$	$\var{a[9]}$
Tasty Hell (g)	$\var{b[0]}$	$\var{b[1]}$	$\var{b[2]}$	$\var{b[3]}$	$\var{b[4]}$	$\var{b[5]}$	$\var{b[6]}$	$\var{b[7]}$	$\var{b[8]}$	$\var{b[9]}$

Give any introductory information the student needs.

			Name	Type	Generated Value

random2	integer	3
a1	integer	75
a2	integer	76
a3	integer	78
a4	integer	80
a5	integer	80
a6	integer	80
a7	integer	81
a8	integer	81
a9	integer	84
a10	integer	85
alist	list	List of 10 items
a	list	List of 10 items
asort	list	List of 10 items

			Name	Type	Generated Value

random	integer	8
b1	integer	57
b2	integer	66
b3	integer	68
b4	integer	70
b5	integer	72
b6	integer	74
b7	integer	78
b8	integer	93
b9	rational	111
b10	rational	111
blist	list	List of 10 items
b	list	List of 10 items
bsort	list	List of 10 items

			Name	Type	Generated Value

meanab	integer	80
meana	rational	80
modea	integer	80
mediana	number	80
meanb	rational	80
modeb	rational	111
medianb	number	73
rangea	number	10
rangeb	rational	54
suma	number	800
sumb	number	800

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(1..3)

JME function reference

Description

Describe what this variable represents, and list any assumptions made about its value.

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Generated value: `integer`

This variable doesn't seem to be used anywhere.

Parts

Part a) Gap-fill
1. Gap Gap 0. Number entry
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2. Gap Gap 1. Number entry
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Part b) Choose one from a list

Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

Using the data above, fill in the following table.

	Sweet Heaven (g)	Tasty Hell (g)
Mean weight	Gap 0.	Gap 1.
Median weight	Gap 2.	Gap 3.
Modal weight	Gap 4.	Gap 5.
Range	Gap 6.	Gap 7.

Advice

We denote Sweet Heaven as $s$ and Tasty Hell as $t$.

a)

We are going to start with completing the column for Sweet Heaven.

First, we need to find the sum of weights of all the scoops:

\[\begin{align} \sum s &= \var{a[0]} + \var{a[1]} + \var{a[2]} + \var{a[3]} + \var{a[4]} + \var{a[5]} + \var{a[6]} + \var{a[7]} + \var{a[8]} + \var{a[9]} \\&= \var{suma} \text{.}
\end{align}\]

The total number of measurements $n$ is $10$.

Therefore the mean is

\[ \begin{align} \overline{s} &= \frac{\sum s}{n} \\[3pt]&= \frac{\var{suma}}{10} \\&= \var{meana} \text{.} \end{align}\]

The median is the middle value. We need to sort the list in order:

Sweet Heaven (g)	$\var{asort[0]}$	$\var{asort[1]}$	$\var{asort[2]}$	$\var{asort[3]}$	$\var{asort[4]}$	$\var{asort[5]}$	$\var{asort[6]}$	$\var{asort[7]}$	$\var{asort[8]}$	$\var{asort[9]}$

There is an even number of responses, so there are two numbers in the middle (5^th and 6^th place). To find the median, we need to find the mean of these two numbers $\var{asort[4]}$ and $\var{asort[5]}$:

\[ \displaystyle \begin{align} \frac{\var{asort[4]} + \var{asort[5]}}{2} &= \frac{\var{asort[4] + asort[5]}}{2} \\&= \var{mediana} \text{.} \end{align}\]

The mode is the value that occurs the most often in the data.

To find a mode, we can look at our sorted list:

Sweet Heaven (g)	$\var{asort[0]}$	$\var{asort[1]}$	$\var{asort[2]}$	$\var{asort[3]}$	$\var{asort[4]}$	$\var{asort[5]}$	$\var{asort[6]}$	$\var{asort[7]}$	$\var{asort[8]}$	$\var{asort[9]}$

We notice that $\var{modea}$ occurs the most times (3) and so $\var{modea}$ is the mode.

The range is the difference between the highest and the lowest value in the data.

To find this, we subtract the lowest value from the highest value:

\[ \var{max(a)} - \var{min(a)} = \var{rangea} \text{.}\]

So the first column is

	Sweet Heaven (g)
Mean weight	$\var{meana}$
Median weight	$\var{mediana}$
Modal weight	$\var{modea}$
Range	$\var{rangea}$

Similarly for Tasty Hell,

\[\begin{align} \sum t &= \var{b[0]} + \var{b[1]} + \var{b[2]} + \var{b[3]} + \var{b[4]} + \var{b[5]} + \var{b[6]} + \var{b[7]} + \var{b[8]} + \var{b[9]} \\&= \var{sumb} \text{.}
\end{align}\]

The total number of measurements $n$ is $10$ again.

Therefore the mean is

\[ \begin{align} \overline{t} &= \frac{\sum t}{n} \\[3pt]&= \frac{\var{sumb}}{10} \\&= \var{meanb} \text{.} \end{align}\]

For median, we sort the list in order:

Tasty Hell (g)	$\var{bsort[0]}$	$\var{bsort[1]}$	$\var{bsort[2]}$	$\var{bsort[3]}$	$\var{bsort[4]}$	$\var{bsort[5]}$	$\var{bsort[6]}$	$\var{bsort[7]}$	$\var{bsort[8]}$	$\var{bsort[9]}$

There is an even number of responses, so there are two numbers in the middle (5^th and 6^th place). We find the mean of these two numbers $\var{bsort[4]}$ and $\var{bsort[5]}$:

\[ \displaystyle \begin{align} \frac{\var{bsort[4]} + \var{bsort[5]}}{2} &= \frac{\var{bsort[4] + bsort[5]}}{2} \\&= \var{medianb} \text{.} \end{align}\]

For mode, we look at our sorted list:

Tasty Hell (g)	$\var{bsort[0]}$	$\var{bsort[1]}$	$\var{bsort[2]}$	$\var{bsort[3]}$	$\var{bsort[4]}$	$\var{bsort[5]}$	$\var{bsort[6]}$	$\var{bsort[7]}$	$\var{bsort[8]}$	$\var{bsort[9]}$

We notice that $\var{modeb}$ occurs the most times (2) and so $\var{modeb}$ is the mode.

To find the range, we subtract the lowest value from the highest value:

\[ \var{max(b)} - \var{min(b)} = \var{rangeb} \text{.}\]

So the complete table is

	Sweet Heaven (g)	Tasty Hell (g)
Mean weight	$\var{meana}$	$\var{meanb}$
Median weight	$\var{mediana}$	$\var{medianb}$
Modal weight	$\var{modea}$	$\var{modeb}$
Range	$\var{rangea}$	$\var{rangeb}$

Let's look at the differences between the two ice cream parlours:

The range of weight of Tasty Hell scoops ($\var{rangeb}$) is far greater than that of Sweet Heaven scoops ($\var{rangea}$).

The mean weight for each shop is $\var{meanab}$. This implies that the scoops are more-or-less the same in both shops. However, looking at the actual values as well as other measures, we can see this is not true, so the mean is not very reliable in this case.

When we compare the medians ($\var{mediana}$ and $\var{medianb}$), we might assume that the scoops are generally lighter in Tasty Hell. This is partly true, but there were some much heavier scoops provided by this shop as well.

Looking at modes ($\var{modea}$ and $\var{modeb}$) can be very misleading, because the modal weight for Tasty Hell is the maximum value at the same time, so it is not a reliable measure of average in this case.

b)

Alice wants her children's ice creams to be very similar.

This is more likely to happen in the shop with a lower range of values.

Comparing the ranges, the range of weight of Sweet Heaven scoops ($\var{rangea}$) is far lower than that of Tasty Hell scoops ($\var{rangeb}$), implying Sweet Heaven is more consistent with their scoops.

Give a worked solution to the whole question.

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Chris Graham 7 years, 9 months ago

Chris Graham commented 7 years, 9 months ago

Stanislav Duris 7 years, 9 months ago

Stanislav Duris 7 years, 9 months ago

Stanislav Duris 7 years, 9 months ago

Elliott Fletcher 7 years, 9 months ago

Elliott Fletcher commented 7 years, 9 months ago

Stanislav Duris commented 7 years, 9 months ago

Stanislav Duris 7 years, 9 months ago

Chris Graham commented 7 years, 9 months ago

Chris Graham 7 years, 9 months ago

Stanislav Duris 7 years, 9 months ago

Stanislav Duris 7 years, 9 months ago

Stanislav Duris 7 years, 9 months ago

Stanislav Duris 7 years, 9 months ago

Error in variable testing condition

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Generated value: `integer`

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a)

b)

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Weight of a scoop in two ice cream parlours

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Chris Graham 7 years, 9 months ago

Chris Graham commented 7 years, 9 months ago

Stanislav Duris 7 years, 9 months ago

Stanislav Duris 7 years, 9 months ago

Stanislav Duris 7 years, 9 months ago

Elliott Fletcher 7 years, 9 months ago

Elliott Fletcher commented 7 years, 9 months ago

Stanislav Duris commented 7 years, 9 months ago

Stanislav Duris 7 years, 9 months ago

Chris Graham commented 7 years, 9 months ago

Chris Graham 7 years, 9 months ago

Stanislav Duris 7 years, 9 months ago

Stanislav Duris 7 years, 9 months ago

Stanislav Duris 7 years, 9 months ago

Stanislav Duris 7 years, 9 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

a)

b)

Generated value: `integer`