Estimate the mean and find the modal class for grouped data

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Xiaodan Leng 5 years, 8 months ago

Created this as a copy of Estimate the mean and find the modal class for grouped data.

Name	Status	Author	Last Modified
Estimate the mean and find the modal class for grouped data	Ready to use	Stanislav Duris	18/09/2020 20:43
Estimate the mean and find the modal class for grouped data	draft	Xiaodan Leng	11/07/2019 01:41

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Question statement

$\var{sum(freq)}$ Mathematics and Statistics students have finished their first year at Newcastle University.

Give any introductory information the student needs.

			Name	Type	Generated Value

freq	list	List of 10 items
freq_midpoint	integer	9870
sixty	integer	59
seventy	integer	55

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
repeat(random(0..8), 2) + random(0..4) + random(6..18) + random(20..40) + sixty + seventy + random(10..40) + random(0..6) + random(0..1)

JME function reference

Description

$0 \leq a \lt 10$	0..8
$10 \leq a \lt 20$	0..7
$20 \leq a \lt 30$	0..4
$30 \leq a \lt 40$	6..18
$40 \leq a \lt 50$	20..50
$50 \leq a \lt 60$	30..60
60 70
$70 \leq a \lt 80$	10..40
$80 \leq a \lt 90$	0..6
$90 \leq a \lt 100$	0..1

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

Can an exam override the value of this variable?

Generated value: `list`

[ 2, 3, 1, 13, 29, 59, 55, 15, 1, 0 ]

← Depends on:

seventy
sixty

→ Used by:

freq_midpoint

Statement
Advice
"Part a)" - prompt
"Part a)" → "Gap 10." - Minimum accepted value
"Part a)" → "Gap 10." - Maximum accepted value
"Part a)" → "Gap 11." - Minimum accepted value
"Part a)" → "Gap 11." - Maximum accepted value
"Part a)" → "Gap 12." - Minimum accepted value
"Part a)" → "Gap 12." - Maximum accepted value
"Part a)" → "Gap 13." - Minimum accepted value
"Part a)" → "Gap 13." - Maximum accepted value
"Part a)" → "Gap 14." - Minimum accepted value
"Part a)" → "Gap 14." - Maximum accepted value
"Part a)" → "Gap 15." - Minimum accepted value
"Part a)" → "Gap 15." - Maximum accepted value
"Part a)" → "Gap 16." - Minimum accepted value
"Part a)" → "Gap 16." - Maximum accepted value
"Part a)" → "Gap 17." - Minimum accepted value
"Part a)" → "Gap 17." - Maximum accepted value
"Part a)" → "Gap 18." - Minimum accepted value
"Part a)" → "Gap 18." - Maximum accepted value
"Part a)" → "Gap 19." - Minimum accepted value
"Part a)" → "Gap 19." - Maximum accepted value
"Part b)" - Minimum accepted value
"Part b)" - Maximum accepted value

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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10. Gap Gap 9. Number entry
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11. Gap Gap 10. Number entry
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12. Gap Gap 11. Number entry
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13. Gap Gap 12. Number entry
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14. Gap Gap 13. Number entry
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15. Gap Gap 14. Number entry
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16. Gap Gap 15. Number entry
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17. Gap Gap 16. Number entry
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18. Gap Gap 17. Number entry
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19. Gap Gap 18. Number entry
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20. Gap Gap 19. Number entry
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21. Gap Gap 20. Number entry
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1. Step Information only
Part b) Number entry
Part c) Choose one from a list

Gap-fill

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

Their stage 1 grades (denoted $a$) have been summarised and grouped into intervals in the frequency table below. Complete the table.

Stage 1 grade $a$	Frequency	Midpoint	Frequency $\times$ Midpoint
$0 \leq a \lt 10$	$\var{freq[0]}$	Gap 0.	Gap 10.
$10 \leq a \lt 20$	$\var{freq[1]}$	Gap 1.	Gap 11.
$20 \leq a \lt 30$	$\var{freq[2]}$	Gap 2.	Gap 12.
$30 \leq a \lt 40$	$\var{freq[3]}$	Gap 3.	Gap 13.
$40 \leq a \lt 50$	$\var{freq[4]}$	Gap 4.	Gap 14.
$50 \leq a \lt 60$	$\var{freq[5]}$	Gap 5.	Gap 15.
$60 \leq a \lt 70$	$\var{freq[6]}$	Gap 6.	Gap 16.
$70 \leq a \lt 80$	$\var{freq[7]}$	Gap 7.	Gap 17.
$80 \leq a \lt 90$	$\var{freq[8]}$	Gap 8.	Gap 18.
$90 \leq a \lt 100$	$\var{freq[9]}$	Gap 9.	Gap 19.
Totals	$\var{sum(freq)}$	$-$	Gap 20.

Advice

a)

We calculate the midpoint by finding the mean of the lower and upper bounds. For example, the midpoint for $40 \leq a \lt 50$ is

\[\begin{align} \displaystyle \frac{40 + 50}{2} &= \frac{90}{2} \\&= 45 \text{.} \end{align}\]

To calculate the final column, we multiply the second and the third columns:

\[\var{freq[4]} \times 45 = \var{45*freq[4]} \text{.} \]

Finally, when we have completed all the other values, we can calculate the total for the last column.

\[\begin{align} \text{Total} &= 5\times\var{freq[0]} + 15\times\var{freq[1]} + 25\times\var{freq[2]} + 35\times\var{freq[3]} + 45\times\var{freq[4]} + 55\times\var{freq[5]} + 65\times\var{freq[6]} + 75\times\var{freq[7]} + 85\times\var{freq[8]} + 95\times\var{freq[9]} \\&= \var{5*freq[0]} + \var{15*freq[1]} + \var{25*freq[2]} + \var{35*freq[3]} + \var{45*freq[4]} + \var{55*freq[5]} + \var{65*freq[6]} + \var{75*freq[7]} + \var{85*freq[8]} + \var{95*freq[9]} \\&= \var{freq_midpoint} \text{.} \end{align}\]

The completed frequency table looks like this:

Stage 1 grade $a$	Frequency	Midpoint	Frequency $\times$ Midpoint
$0 \leq a \lt 10$	$\var{freq[0]}$	$5$	$\var{5*freq[0]}$
$10 \leq a \lt 20$	$\var{freq[1]}$	$15$	$\var{15*freq[1]}$
$20 \leq a \lt 30$	$\var{freq[2]}$	$25$	$\var{25*freq[2]}$
$30 \leq a \lt 40$	$\var{freq[3]}$	$35$	$\var{35*freq[3]}$
$40 \leq a \lt 50$	$\var{freq[4]}$	$45$	$\var{45*freq[4]}$
$50 \leq a \lt 60$	$\var{freq[5]}$	$55$	$\var{55*freq[5]}$
$60 \leq a \lt 70$	$\var{freq[6]}$	$65$	$\var{65*freq[6]}$
$70 \leq a \lt 80$	$\var{freq[7]}$	$75$	$\var{75*freq[7]}$
$80 \leq a \lt 90$	$\var{freq[8]}$	$85$	$\var{85*freq[8]}$
$90 \leq a \lt 100$	$\var{freq[9]}$	$95$	$\var{95*freq[9]}$
Totals	$\var{sum(freq)}$		$\var{freq_midpoint}$

b)

We don't know exactly how much each student got, so we have to assume all students got the midpoint of their group. To estimate the sum of all student grades in a particular interval, we can multiply the interval's midpoint by its frequency.

The sum of all the grades in the class is the sum of the estimates for each interval. We've already calculated this in the table above.

So the estimate for the mean, $\bar{a}$, is as follows:

\begin{align}
\bar{a} &\approx \frac{\var{freq_midpoint}}{\var{sum(freq)}} \\
&= \var{freq_midpoint/sum(freq)} \text{.}
\end{align}

Rounding the mean to 2 decimal places, we get $\var{precround(freq_midpoint/sum(freq),2)}$.

c)

The modal class is the interval with the highest frequency. In this case, the interval $50 \leq a \lt 60$ is the modal class.

Give a worked solution to the whole question.

Use this tab to check that this question works as expected.

There was an error which means the tests can't run:

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Choose one from a list
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This question is used in the following exam:

Apprentices Year 2 Descriptive Statistics Quiz by Aoife O'Brien in Aoife's workspace.

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Estimate the mean and find the modal class for grouped data

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

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Checkpoint description

Xiaodan Leng 5 years, 8 months ago

Error in variable testing condition

Error

Warning

Generated value: `list`

Parts

Gap-fill

a)

b)

c)

Loading...

Error

Estimate the mean and find the modal class for grouped data

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Xiaodan Leng 5 years, 8 months ago

Error in variable testing condition

Error

Warning

Generated value: list

Parts

Gap-fill

a)

b)

c)

Generated value: `list`