Michael's copy of Weight of a scoop in two ice cream parlours

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Michael Proudman 6 years, 5 months ago

Created this as a copy of Weight of a scoop in two ice cream parlours.

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

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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	1
a1	integer	87
a2	integer	88
a3	integer	88
a4	integer	90
a5	integer	90
a6	integer	90
a7	integer	91
a8	integer	91
a9	integer	92
a10	integer	93
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	67
b2	integer	76
b3	integer	78
b4	integer	80
b5	integer	82
b6	integer	84
b7	integer	88
b8	integer	103
b9	rational	121
b10	rational	121
blist	list	List of 10 items
b	list	List of 10 items
bsort	list	List of 10 items

			Name	Type	Generated Value

meanab	integer	90
meana	rational	90
modea	integer	90
mediana	number	90
meanb	rational	90
modeb	rational	121
medianb	number	83
rangea	number	6
rangeb	rational	54
suma	number	900
sumb	number	900

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.

Can an exam override the value of this variable?

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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3. Gap Gap 2. Number entry
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4. Gap Gap 3. Number entry
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5. Gap Gap 4. Number entry
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6. Gap Gap 5. 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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There was an error which means the tests can't run:

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This question is used in the following exam:

Michael's copy of Harry's copy of Measures of central tendency and spread-original without context by Michael Proudman in .Statistics.

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Michael's copy of Weight of a scoop in two ice cream parlours

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Michael Proudman 6 years, 5 months ago

Error in variable testing condition

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Warning

Generated value: `integer`

Parts

Gap-fill

a)

b)

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Michael's copy of Weight of a scoop in two ice cream parlours

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Michael Proudman 6 years, 5 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

a)

b)

Generated value: `integer`