12: Statistics - Borehole - Numbas at mathcentre.ac.uk

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Nasir Firoz Khan

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John Hodkinson	said	Ready to use	8 years, 8 months ago
Nasir Firoz Khan	said	Needs to be tested	8 years, 9 months ago

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Jinhua Mathias 8 years, 1 month ago

Published this.

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John Hodkinson 8 years, 8 months ago

Gave some feedback: Ready to use

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Nasir Firoz Khan 8 years, 9 months ago

Gave some feedback: Needs to be tested

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Nasir Firoz Khan 8 years, 9 months ago

Created this.

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Question statement

The following table shows the thickness, in mm, of a sample of the strata found in 2 separate boreholes.

Borehole A (mm)	Borehole B (mm)
{ia[0]}	{ib[0]}
{ia[1]}	{ib[1]}
{ia[2]}	{ib[2]}
{ia[3]}	{ib[3]}
{ia[4]}	{ib[4]}
{ia[5]}	{ib[5]}
{ia[6]}	{ib[6]}
{ia[7]}	{ib[7]}
{ia[8]}	{ib[8]}
{ia[9]}	{ib[9]}

Useful formulae

For $N$ measurements, $x_i$ , $i = 1,2,3,...,N$

mean, $\bar x = \frac{1}{N}\sum\limits_{i=1}^{N}x_i$

sample variance, $s^2 = \frac{1}{N}\big(\sum\limits_{i=1}^{N}(x_i - \bar x)^2\big) = \overline {x^2} - \bar x^2$

unbiased estimate of population variance, $\hat s^2 = \frac{N}{N-1} s^2$

standard deviation is the square root of variance.

standard error in the mean, $SE = \frac{\hat s}{\sqrt{N}}$

Complete the following:

Give any introductory information the student needs.

			Name	Type	Generated Value

ia	list	List of 10 items
ib	list	List of 10 items
meda	number	29.5
medb	number	24.5
sampvara	number	3.25
sampvarb	number	2.89
meana	number	30.5
meanb	number	24.9
popmeana	integer	31
popmeanb	integer	26
sea	number	0.6009252126
seb	number	0.5666666667

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
repeat(round(normalsample(popmeana,3)),10)

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: `list`

[ 29, 32, 29, 29, 29, 34, 29, 31, 33, 30 ]

← Depends on:

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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1. Step Information only
Part b) Gap-fill
1. Gap Gap 0. Number entry
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2. Gap Gap 1. Number entry
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Part e) Gap-fill
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Gap-fill

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

Calculate the mean thickness for each sample.

Mean thickness of sample in borehole A: Gap 0. mm

Mean thickness of sample in borehole B: Gap 1. mm

Advice

a) We use the formula for the mean. In our case we have $N$ = 10 since we have 10 measurements for each borehole. We find the total for Borehole A to be {meana*10} and the total for Borehole B to be {meanb*10}. All that remains is to divide each by $N$ = 10.

b) We first order each list of measurements (ascending or descending order will do!).

Borehole A measurements = {ordered(ia)}

Borehole B measurements = {ordered(ib)}

Since we have 10 measurements, the medians lie between the fifth and sixth values in our ordered lists. To calculate them we simply take the average of those respective values:

Median of A = $\frac{\var{ordered(ia)[4]} + \var{ordered(ia)[5]}}{2} = \var{meda}$

Median of B = $\frac{\var{ordered(ib)[4]} + \var{ordered(ib)[5]}}{2} = \var{medb}$

c) We use either form of the formula for the sample variance. Here we will use the simpler form which requires us to first calculate the sum of squared measurements for each sample. (Make sure you understand and can apply these useful formulae before the exam!)

Sample A variance = $\overline {x^2 }- \bar x^2 = \frac{\var{sumsqrd(ia)}}{10} - \var{meana}^2 = \var{sampvara}$

Sample B variance = $\overline {x^2 }- \bar x^2 = \frac{\var{sumsqrd(ib)}}{10} - \var{meanb}^2 = \var{sampvarb}$

d) We note that we are asked for the standard deviation, not the variance, so we will need to take a square root at some point. We want to obtain $\hat s$, so we take the square root of the given formula for the unbiased estimate of population variance: $\hat s =\sqrt{ s^2\frac{N}{N-1}}$. We now plug in our previously calculated values of the sample variance, $s^2$, and round to 3 decimal places:

$\hat s_A = \sqrt{\var{sampvara}\times\frac{10}{9}} = \var{sqrt({sampvara}*10/9)} =\var{precround(sqrt({sampvara}*10/9),3)} $

$\hat s_B = \sqrt{\var{sampvarb}\times\frac{10}{9}} = \var{sqrt({sampvarb}*10/9)}=\var{precround(sqrt({sampvarb}*10/9),3)} $

e) We use the formula for the standard error in the mean. We calculated the values of $\hat s$ in part d), so all that is left to do is to divide these values by $\sqrt{10}$.

$SE_A$= $\var{sqrt({sampvara}*10/90)}= \var{precround(sqrt({sampvara}*10/90),3)}$

$SE_B$= $\var{sqrt({sampvarb}*10/90)}= \var{precround(sqrt({sampvarb}*10/90),3)}$

(You might have a slightly different answer if you have used the rounded value from part d). It is good practice to use the unrounded values in later calculations unless directed otherwise. You can make this easier by storing the previous answers in your calculator's memory. If you do not know how to do this, please follow the link to see a 30-second tutorial: https://vimeo.com/101615630)

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12: Statistics - Borehole

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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Jinhua Mathias 8 years, 1 month ago

John Hodkinson 8 years, 8 months ago

Nasir Firoz Khan 8 years, 9 months ago

Nasir Firoz Khan 8 years, 9 months ago

Error in variable testing condition

Error

Warning

Generated value: `list`

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Gap-fill

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12: Statistics - Borehole

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Jinhua Mathias 8 years, 1 month ago

John Hodkinson 8 years, 8 months ago

Nasir Firoz Khan 8 years, 9 months ago

Nasir Firoz Khan 8 years, 9 months ago

Error in variable testing condition

Error

Warning

Generated value: list

Parts

Gap-fill

Generated value: `list`