Trinity College, Dublin
Generic Skills Programme
Statistics for Research Students
Laboratory 3:
Control Charts and Statistical significance
To complete the laboratory exercise, work your way through this handout, which is self
contained and self explanatory. Work in pairs (two per machine), and learn from each other.
Keep separate logs of your work. The tutor is available to help with technicalities and discuss
substantive issues if necessary.
Invitations to consider the results of Minitab analysis and their statistical and substantive
interpretations are printed in italics. Take some time for this; consult your neighbour or
tutor. Enter your responses in a Word document, as if draft contributions to a report on
the experiment and its analysis.
Topics:
1.
2.
3.
4.
Control charts; preliminary analysis and set-up
Simulating sampling distributions
One-sample significance tests and control charts
Application to paired comparisons
Control charts are used in two modes, ongoing monitoring of process measurements and
analysis of historical process data. The latter may be used at the initial stage of setting up
ongoing process monitoring, with the results of the analysis being used to determine the
parameters of the control chart for ongoing monitoring. Historical analysis should also be used
on a regular basis to assess overall recent process performance, with a view to identifying
actual or potential problems and thereby setting up opportunities for improvement.
Part 1 of today's Laboratory is based on an example of the use of historical analysis of regularly
sampled process data to learn about process variation and to determine the set up for ongoing
monitoring.
The performance of the control chart in ongoing process monitoring may be assessed
theoretically through the sampling distribution of the statistic of interest. Alternatively,
simulation of ongoing process performance may be used as a basis on which to assess the
performance of the control chart, effectively simulating the sampling distribution and assessing
its characteristics through observation. This is undertaken in Part 2.
In Part 3, the correspondence between the logic of control chart monitoring and statistical
significance testing is explored and the use of statistical significance testing in historical control
chart analysis is demonstrated. There are theoretical assumptions concerning the process that
generated the data regarding Normality, homogeneity and independence of the sampled data.
Diagnostic analysis is used to assess these assumptions.
In Part 4, the use of the simple one-sample test which is the basis for control chart monitoring is
applied in an experimental setting, where the "matched pairs" design is used as a highly
effective approach to investigating the effect of changing a factor thought to influence a
process. The test is applied to the difference resulting from the paired comparisons arising
from the implementation of the design.
Trinity College, Dublin
Generic Skills Programme
Statistics for Research Students
Laboratory 3
Learning Objectives:
Be able to
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use Minitab to create control charts from subgrouped data
interpret the resulting charts in context and decide on relevant action
use Minitab to create control charts for subsets of the data
explain the basis for choosing control chart limits in terms of chart behaviour in
repeated sampling
simulate repeated samples and observe control chart behaviour
create suitable graphical summaries of control chart behaviour to display the effect
of increasing sample size
use the Minitab graph editor to improve display intelligibility
provide informative interpretive comments on the results of the graphical analysis,
with specific reference to standard error
use Minitab to reconfigure subgrouped data in a single column as individual
subgroups in separate columns
use Minitab to apply a one sample Z test to the data in the individual subgroups
calculate a one sample Z test using the standard formula
verify the correspondence between the Minitab Z test, the formula based Z test and
the Xbar control chart test, with specific reference to correspondence between
critical values and correspondence between significance levels
use Minitab to group relevant subgroups and use a t test to test a relevant
hypothesis separately for each group
verify the numerical results of the tests by calculation using the relevant formula
provide formal reports on the results of the t tests
explain the effects of varying mean, standard deviation and sample size on the result
of a t test
calculate appropriate residuals, produce appropriate graphs and use appropriate
graphical analysis to check the assumptions underlying the validity of control chart
analysis
produce, edit and interpret profile plots of paired sample measurements and
corresponding differences
produce, edit and interpret a scatter plot of differences of pairs versus averages of
pairs and a Normal plot of differences
use Minitab to test the statistical significance of the mean difference, verify the
calculations and make a formal report on the result of the test.
page 2
Trinity College, Dublin
Generic Skills Programme
1.
Statistics for Research Students
Laboratory 3
Control charts; preliminary analysis and set-up
As part of a preliminary study of the bill-paying behaviour of its customers, a mail order
company collected data on the number of days to collection of accounts receivable. Rather
than observe every account, the first five accounts received on Monday morning each week
were taken as a sample of the process. After 30 weeks, the data were reviewed. The first step
was to assess the stability of the process over the 30 week period. For this purpose, X-bar and
s charts were made using Minitab.
The data are available in the Days to Payment data set in the GenericSkillsData folder. Copy
the data to Minitab and create control charts as follows:
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from the Stat menu, select Control Charts, then Variables Charts for Subgroups, then
Xbar-s Chart,
in the top right window of the dialog box, select "Observations for a subgroup are in one
row of columns":
tab to the next window, then select 'Sample 1'-'Sample 5' from the variables list,
click the Xbar-S Options button, click the Estimate tab, uncheck the "Use unbiasing
constant" box1,
click OK, OK.
Discuss the stability of the days to payment process with regard to
(a)
(b)
level
spread
What is your next step in the analysis?
As the variation in Days to Payment process appeared stable during the last 22 weeks, it was
decided to monitor the process on an ongoing basis, using control limits based on the recent
data. To establish these, charts based on the recent data were produced, using the following
procedure:
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from the Edit menu, select Edit Last Dialog (or press Ctrl+E),
click on Data Options,
check "Specify rows to include", then "Rows that match", click "Condition",
enter Week > 8 as the Condition,
click OK, OK, OK.
Confirm that the control limits for the Xbar chart conform to the formula CL  3/n.
List values for centre lines and control limits to use for further process monitoring.
Use sensible rounding.
Division by n – 1 in the denominator of s eliminates the bias in s 2 as an estimate of 2, but this does not
apply to s; division by n – 1 reduces but does not eliminate the bias in s as an estimate of  A further
adjustment is available to "unbias" s. Formulas for doing this are rather complicated, but have been
incorporated in Minitab. Minitab offers an option of using such a formula in constructing the s chart, but
appears to use a different unbiasing adjustment in calculating the control limits for the Xbar chart. The
unbiasing option is ignored here, to make both charts consistent.
1
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Trinity College, Dublin
Generic Skills Programme
2.
Statistics for Research Students
Laboratory 3
Simulating sampling distributions
The basis for interpreting control charts is the notion of sampling distribution. In the context of
process sampling, this requires us to think of continuing sampling indefinitely. Then, if the
process is in control, very few samples will give a value of X-bar outside the control limits; in the
long run, the Normal model indicates that less than 0.3% of X-bar values will be outside the
control limits. If, in the short term, we observe such a point, we are inclined to believe that it
reflects an assignable cause of variation rather then chance causes associated with the Normal
model. Put another way, assuming the process is in control makes observing a point outside
the control limits improbable, so that observing such a point makes the in control assumption
implausible.
We can verify these properties of the X-bar chart by simulation, as follows:
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from the File menu, select New, then Minitab Project,
click No, OK.
generate 30 rows of Normal data in columns C1-C5, with mean and standard deviation
set to the values you prescribed at the end of Part 1,
name the columns Sample 1 – Sample 5
use Calc / Row Statistics to calculate the means of each row of five values, each row
representing a simulated sample of 5, store in C6,
name C6 as Xbar
count the number of values of Xbar outside the control limits you prescribed at the end
of Part 1,
Discuss the result of this simulation.
For a more informative view of the results of the simulation,
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make a histogram of the 30 sample means, with fitted Normal curve.
Comment on the result.
The next step is to increase the size of the simulation to 300 samples and then 3,000 samples.
Counting out of control values by hand becomes impractical. Minitab can be made to count the
values outside the control limits, as follows:
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from the Calc menu, select Calculator,
Store result in variable C7,
from the Function menu, select the If function,
as the test, enter C6 < LCL OR C6 > UCL, where LCL and UCL are the control limit
values you prescribed for the Xbar chart at the end of Part 1,
substitute 1 for value_if_true,
substitute 0 for value_if_false,
check the Assign as a formula box,
click OK,
name C7 as OutOfControl,
from the Calc menu, select Calculator,
Store result in variable C8,
clear the Expression window and, from the Function menu, select the Sum function,
enter C8 as the number in the Sum function,
check the Assign as a formula box,
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Trinity College, Dublin
Generic Skills Programme
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Statistics for Research Students
Laboratory 3
click OK,
name C8 as Count
Check the result.
Next, repeat the simulation with 300 samples (rows), then 3,000 samples, in C1 – C5, and
make a histogram of Xbar in each case. Note that the dialog windows for each command need
no change, apart from the number of simulations initially. This means that you can go straight
to the final OK button. Also, the out-of-control count is automatic, as a result of checking the
"Assign as a formula" box.
Compare the results of the simulations.
What out-of-control count did you get in each case? Keep a record.
What did you expect?
How do the histograms compare?
Repeat the simulation with 3000 repetitions 4 times, recording the out-of-control count each
time.
In the data sheet circulated in class, enter the out-of-control counts for all 5 simulations based
on 3,000 repetitions, for later discussion.
Simulating the effect of increasing sample size
Increasing the size of the samples on which the control charts are based provides more
information about the process being sampled and, therefore, should increase the precision of
our conclusions regarding the process being sampled. Simulation can be used to study this
effect. Here, we will compare histograms of means of 1000 samples of sizes 1, 5, 10, 20, 40,
as follows:
 generate into C1-C40 1000 rows of Normal data with mean and standard deviation set
to the values you prescribed at the end of Part 1,
 use the Row Statistics command to calculate the means of C1-C5, C1-C10, C1-C20 and
C1-C40, respectively, and store the results in C41, C42, C43 and C44, respectively,
 name C40 as "n = 1", C41 as "n = 5", C42 as "n = 10", C43 as "n = 20", C44 as "n = 40",
 make histograms of C40-C44 with fitted Normal curves,
 select Scale and delete all Y scale axes and ticks and the high X scale axis,
 select Multiple Graphs, check On separate graphs and Same X scale for graphs,
 click OK.
The histograms are cascaded, with the last (n = 40) showing on top. Successively show the
other histograms by clicking on them in turn, back to the first (n = 1).
Optionally, use the layout tool to improve comparability:
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from the Editor menu, select the Layout tool,
set Rows = 5, Columns = 1,
transfer Histogram of n = 1 to the first cell of the layout (clearing other histograms first if
necessary), then successively transfer the histograms for n = 5, n = 10, n = 20, n = 40,
click Finish,
page 5
Trinity College, Dublin
Generic Skills Programme
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Statistics for Research Students
Laboratory 3
use Edit Graph Regions to set Graph Size, setting Width to be half Height and ensuring
that Zoomed Size is set to Fit Window,
drag the Layout window by its Title bar, as high as possible,
resize the Layout window by dragging its bottom right corner as low as possible, while
making it fit the graph region (no grey area showing),
select and delete all figure titles and Y axis titles, leaving X axis titles (n = 1, etc.) as they
are.
Comment on the effect of increasing sample size, referring to
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histogram spread,
values of StDev,
values of Mean.
Compare
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the values of StDev for n = 5 and n = 20,
the values of StDev for n = 10 and n = 40,
noting that the larger sample size is 4 times the smaller in both cases.
Explain your comparisons in terms of the formula /n for the standard error of the
sample mean
3
One-sample significance tests and control charts
The logic of control chart analysis closely parallels that of statistical significance tests. The two
activities are identical in the case that the control chart is being used to monitor the adherence
of a process to a standard or a specification2. To illustrate the latter, data sampled from a
mechanical process to manufacture a metal clip such as that illustrated below will be studied.
The critical measurement being monitored is the "clip gap", indicated by the arrowed line.
According to the engineering specification, the clip gap is required to be between 50 mm and 90
mm3.
2
It is interesting to note that the logic of control charts, as developed by Shewhart, and that of statistical
significance tests, as developed by Fisher, emerged more or less contemporaneously. Shewhart's work
first appeared as an internal memorandum at the Bell Telephone Laboratories, New Jersey, USA, in 1924,
while Fisher's ideas, developed at the Rothamstead agricultural research station, north of London, were
published in his monograph Statistical Methods for Research Workers in 1925.
3 Clearly, the data recorded here have been "coded"; a permitted variation of over 50% of a typical value is
unrealistic. The original clip gap measurements must have been considerably larger, perhaps of the order
of 1,050 mm to 1,090 mm. Unfortunately, the source for these data does not say.
page 6
Trinity College, Dublin
Generic Skills Programme
Statistics for Research Students
Laboratory 3
The process is sampled every two hours, with 5 clips being sampled and measured. Data for
25 successive samples of 5 measurements may be found in the Clip gaps dataset in the
GenericSkillsData folder and are tabled below.
Sample
1
65
70
65
65
85
2
75
85
75
85
65
3
75
80
80
70
75
4
60
70
70
75
65
5
70
75
65
85
80
6
60
75
75
85
70
7
75
80
65
75
70
8
60
70
80
75
75
9
65
80
85
85
75
10
60
70
60
80
65
11
80
75
90
50
80
12
85
75
85
65
70
Sample
13
70
70
75
75
70
14
65
70
85
75
60
15
90
80
80
75
85
16
75
80
75
80
65
17
75
85
70
80
70
18
75
70
60
70
60
19
65
65
85
65
70
20
60
60
65
60
65
21
50
55
65
80
80
22
60
80
65
65
75
23
80
65
75
65
65
24
65
60
65
60
70
25
65
70
70
60
65
Given the engineering specifications, a plausible target mean value for this process is 70 mm.
Historical data suggest a standard deviation of 8. With this information, a control chart for
monitoring the process may be produced as follows:
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from the File menu, select New, then Minitab Project,
click No, OK.
copy the data from the GenericSkillsData folder to Minitab,
from the Stat menu, select Control Charts, then Variables Charts for Subgroups, then
Xbar-s Chart,
in the top right window of the dialog box, select "All observations for a chart are in one
column":
tab to the next window, then select Clip gap from the variables list,
enter 5 as subgroup size,
click the Xbar-S Options button,
click the Parameters tab and enter 70 for the mean and 8 for the standard deviation,
click the Estimate tab and ensure the "Use unbiasing constant" box is unchecked,
click OK, OK.
In real time, the chart you have produced would build up sample by sample, one every two
hours. Here, all samples have been charted at once.
You can monitor the process, sample by sample, by carrying out tests of significance on the
successive samples. First, put the separate samples in separate columns:
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from the Data menu, select Unstack Columns,
click in the Unstack the data in: window and select the Clip gap column,
click the Using subscripts in: window and select Sample,
select After last column in use and check Name the columns containing the unstacked
data,
click OK.
Next, test the hypothesis that the process is on target using one of the samples represented in
the Xbar chart, say Sample 5:
page 7
Trinity College, Dublin
Generic Skills Programme
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Statistics for Research Students
Laboratory 3
from the Stat menu, select Basic Statistics, then 1-Sample Z,
click in the Samples in columns: window and select Clip gap_5,
enter 8 for Standard deviation, check Perform hypothesis test, enter 70 for Hypothesised
mean:,
click OK.
What is the value of Z?
What is the value of P?
What conclusion do you draw?
Correspondence between control chart test and significance test
You have used a Z test here to emulate the control chart test. The formula for the X chart
control limits is
CL  3/n, that is, 70  3×/5
The formula for Z is
Z
X   0 X  70

.

8
n
5
Note that the relevant sampling distribution for Z is the standard Normal distribution, as the
value for  is assumed to be the known value of 8.
Verify the values of the control limits as shown on the X chart.
Verify the value of the Z statistic as shown in the Session window.
Verify the correspondence between the control chart test and the Z test.
What critical value for Z is needed to ensure the correspondence?
What is the significance level corresponding to this critical value?4 Use the Normal
table and / or the Minitab Normal cumulative distribution function (Calc menu).
Check the comparison of the P-value shown in the Session window with the
significance level of the Z test and verify the conclusion of the Z test.
Repeat the Z test for Sample 15; repeat the verification exercise.
4
The conventional significance level for a Z test is 0.05, corresponding to a critical value of 2 (or 1.96, to
be spuriously accurate). Shewhart chose 3 as the critical value for control charts to avoid too many false
alarms that might arise with frequent sampling in an industrial setting. See Stuart (2003, pages 167-8) for
further discussion of the choice of significance levels and critical values.
page 8
Trinity College, Dublin
Generic Skills Programme
Statistics for Research Students
Laboratory 3
An extension of the control chart Z test
A review of the Xbar chart suggests a process shift after Sample 17; most of the points up to
sample 17 are above the centre line, all the points after Sample 17 are below the centre line5.
This is reinforced by re-analysing the (now historical) data in separate subsets. In Minitab, this
is achieved by using "Staging", as follows:
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from the Stat menu, go to Xbar-s Chart, note existing set-up,
click the Xbar-S Options button,
click the Parameters tab and clear 70 as mean and 8 as standard deviation,
click the Estimate tab and ensure the "Use unbiasing constant" box is unchecked,
click the Stages tab, enter Sample as the variable to start a new stage and 18 as the
value to start a new stage,
click OK, OK.
Interpret the revised charts.
Comment on the non-significance of Sample 15.
It appears that the process is stable within each subset but centred at a mean value that may
be different from 70. While the individual samples may not have suggested this (apart from
Sample 15), we are now in a position to test the "on-target" hypothesis within each subset
based on combined samples of historical data. These combined samples are much larger than
the size 5 samples used in the ongoing monitoring.
Implement these tests as follows, first using the Stack command to combine the individual
samples into the relevant subsets:
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from the Data menu, select Stack, then Columns,
in the "Stack the following columns:" window, select Clip gap_1 to Clip gap _17,
select "Column of current worksheet:" and enter C28,
click OK,
in the data sheet window, name C28 as "Before",
stack the remaining samples in C29 and name as "After",
from the Stat menu, choose Basic Statistics, then 1-Sample t,
click in the "Samples in columns:" window and select Before and After,
check Perform hypothesis test and enter 70 as the Hypothesised mean,
click OK.
The column names "Before" and "After" refer to before and after the process adjustment
referred to in Footnote 5.
What is the result of the t-test applied to the "Before" data? Report formally in terms
of
Null hypothesis
Test statistic
Calculated value
Critical value
Comparison
Conclusion
5
A review of the operation of the process indicated that a new roll of the steel raw material from which the
clips were made had been introduced after the time that sample 17 had been taken. The tension in the
steel roll was known to affect the clip gap.
page 9
Trinity College, Dublin
Generic Skills Programme
Statistics for Research Students
Laboratory 3
Confirm the values of N, SE Mean and T for the "Before" data. How many degrees
of freedom did you associate with T?
Repeat the above analysis for the "After" data.
Explain why the deviation of the "Before" data from 70 appears considerably more
significant than that of the "After" data. What factors influence this difference
between the two tests?
Diagnostic analysis
Before accepting the results of the two significance tests as reported (and, indeed, the
application of the control chart methodology), we need to be assured that the assumptions
underlying the analysis were valid. A range of diagnostics is available for this purpose.
To assess the homogeneity of spread, the s chart may be used.
Review the s charts you constructed earlier and comment on the "constant standard
deviation" assumption.
Independence may be assessed by making time series charts, as follows:
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from the Graph menu, select Time Series Plot, then Simple,
select Before as the Series variable,
click OK,
repeat for the "After" variable.
Describe the variation pattern(s) you see in these data.
Are there any patterns that would undermine an assumption of pure chance
variation?
The assumption of Normality may be checked using Normal diagnostic plots:

make Normal diagnostic plots of the "Before" and "After" variables.
Describe the variation pattern(s) you see in these data.
Are there any patterns that would undermine an assumption of Normality?
Diagnostic analysis using Residuals
These diagnostic analyses may be applied to the entire data set by first calculating "residuals",
that is, by subtracting the appropriate subset mean from the values in each subset so that both
subsets, so adjusted, have a common mean of 0, and then recombining the subsets into one
again, as follows:
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from the Calc menu, select Calculator,
Store result in variable C30,
in the Expression window, select the "Before" variable, enter a minus, then select
function MEAN and select the "Before" variable again, click OK,
repeat for the "After" variable, storing the result in C31,
page 10
Trinity College, Dublin
Generic Skills Programme
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Statistics for Research Students
Laboratory 3
stack columns C30 and C31 in C32,
make a time series plot and a Normal plot of C32.
Describe the variation pattern(s) you see in these data.
Are there any patterns that would undermine an assumption of pure chance variation
or of Normality?
4.
Application to Paired Comparisons
Paired comparisons arise from a very effective form of experimental design, the matched pairs
design, for evaluating the effect of changing a factor of interest. Testing the statistical
significance of the factor effect in such a design reduces to a one sample test. This set up is
illustrated here via a study designed to assess the effect of cigarette smoking on the
aggregation of platelets in the blood, a key factor in the formation of blood clots.
Blood was taken from 11 subjects before and after they smoked a cigarette and the percentage
of platelets aggregated in each sample was measured, with the following results.
Subject
1
2
3
4
5
6
7
8
9
10
11
Before
25
25
27
44
30
67
53
53
52
60
28
After
27
29
37
56
46
82
57
80
61
59
43
A more informative view is afforded by a profile plot, which allows a comparison of the Before
and After profiles across all subjects. The Line Plot command in the Graph menu may be used
to produce this, as follows:
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from the File menu, select New, then Minitab Project,
click No, OK.
copy the data from the Platelets dataset in the GenericSkillsData folder to Minitab,
from the Graph menu, select Line Plot, then Series in Rows or Columns (With Symbols),
click OK,
select Before and After as the Graph variables,
select Subject as the Label column,
select "Each column forms a series",
click OK,
double click on the chart title and change "Line" to "Profile",
double click the Y axis title and rename it as "Platelets, per cent".
Comment on the variation pattern in the graph, or lack thereof, with particular
attention to correspondences between pairs of measurements on subjects.
Given the correspondence pattern, this graph may be enhanced by adding the subject
differences:
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use the Minitab Calculator to enter the differences, After minus Before, in the next
available column, name as Difference,
repeat the Line Plot command, this time adding Difference to the list of variables to be
plotted,
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Trinity College, Dublin
Generic Skills Programme
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Statistics for Research Students
Laboratory 3
click OK.
Comment on the variation pattern.
How does the range of variation of the differences relate to the ranges of variation of
the measurements?
How does the size of the differences relate to the size of the measurements?
To answer the last question, a more powerful view is provided by a graph of the subject
differences against the subject averages, constructed as follows:
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use the Row Means command from the Calc menu to enter the subject (row) means of
Before and After in the next available column, name as Means,
from the Graph menu, select Scatterplot, then Simple,
select Difference as the Y variable, Means as the X variable,
click OK.
Comment on the variation pattern.
How does the size of the differences relate to the size of the measurements?
How do you regard the largest difference?
A Normal diagnostic plot provides a better assessment of the size of the largest difference.

Produce a Normal plot of the differences.
Comment on the Normality of the differences and on the exceptional (or otherwise)
status of the largest difference.
Note the similarity of the next three largest differences.
Having investigated the validity of the standard assumptions for testing these data, admittedly
with a rather small sample, we now proceed to test the significance of the differences between
After and Before, as follows:
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use the Basic Statistics command from the Stat menu to implement a one-sample t-test
of the significance of the differences from 0,
alternatively (or in addition), use the paired t option.
Confirm the values of SE Mean, T and P.
Make a formal report of the result, as previously (see page 9).
page 12
Trinity College, Dublin
Generic Skills Programme
Statistics for Research Students
Laboratory 3
Conclusion
This concludes Laboratory 3. The learning objectives listed at the outset are reproduced here.
Check them individually and ensure that you have achieved each one; seek help from the Tutor
if necessary.
Learning Objectives:
Be able to
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use Minitab to create control charts from subgrouped data
interpret the resulting charts in context and decide on relevant action
use Minitab to create control charts for subsets of the data
explain the basis for choosing control chart limits in terms of chart behaviour in
repeated sampling
simulate repeated samples and observe control chart behaviour
create suitable graphical summaries of control chart behaviour to display the effect
of increasing sample size
use the Minitab graph editor to improve display intelligibility
provide informative interpretive comments on the results of the graphical analysis,
with specific reference to standard error
use Minitab to reconfigure subgrouped data in a single column as individual
subgroups in separate columns
use Minitab to apply a one sample Z test to the data in the individual subgroups
calculate a one sample Z test using the standard formula
verify the correspondence between the Minitab Z test, the formula based Z test and
the Xbar control chart test, with specific reference to correspondence between
critical values and correspondence between significance levels
use Minitab to group relevant subgroups and use a t test to test a relevant
hypothesis separately for each group
verify the numerical results of the tests by calculation using the relevant formula
provide formal reports on the results of the t tests
explain the effects of varying mean, standard deviation and sample size on the result
of a t test
calculate appropriate residuals, produce appropriate graphs and use appropriate
graphical analysis to check the assumptions underlying the validity of control chart
analysis
produce, edit and interpret profile plots of paired sample measurements and
corresponding differences
produce, edit and interpret a scatter plot of differences of pairs versus averages of
pairs and a Normal plot of differences
use Minitab to test the statistical significance of the mean difference, verify the
calculations and make a formal report on the result of the test.
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