Normal Probability Plot & Confidence Interval

Simulations using MINITAB

In this session we shall use Minitab

- create a simulated set of data for use in Minitab,
- display a graphical summary of the data,
- produce a normal probability plot for the data,
- construct a one-sample confidence interval on the mean,
- use Excel
^{®}to produce a normal probability plot for the data.

We shall simulate the creation of a The standard exponential distribution We shall reproduce the lamp life Start Minitab. Click on the menu item " Move to " On the new pop-up menu, |

In the new dialog box, Enter Type Enter Leave the Threshold at 0 and Click on the " |

Type a new name for column
As lifetimes are often random quantities that The column Note that this is a set of random data. |

As we have done before, we can use Minitab to gain a first impression of the distribution of our newly-generated data. Click on the menu item
" Move to " Click on " |

In the left pane of the dialog box, double click on " click on the " |

In the new dialog box, Check the boxes " Click on the " Back in the " |

Two graphs then appear in the output pane:

The evidence for positive skew is overwhelming.

The histogram shows a long right tail and no left tail.

The boxplot shows the upper quartile to be further away from
the median than the lower quartile.

The boxplot’s upper whisker is much longer than its lower
whisker.

There are several outliers, all on the positive side.

From the top of the output pane,

The mean is much greater than the median.

[Note also that the values of sample mean and sample standard deviation are consistent with equality of the population mean and population standard deviation, which is true of any exponential distribution.]

Clearly the normal distribution does not fit these data.

Click on the menu item
" Click on
" |

Accept the default "Single" Just click on the " |

In the left pane of the dialog box,

Double click on "`C1 Lifetime`

"
to select it into the "`Graph variables`

" pane, then

Click on the "`Labels...`

" button.

Provide a more meaningful title for the normal probability plot.

Then click on the "`OK`

" button.

Back in the "`Probability Plot`

"
window, click on the "`OK`

" button.

The following normal probability plot (or something very like it) then appears in a new tab of the output pane:

A random sample of size 100, drawn from a normal distribution,
will have all (or nearly all) of its points near the straight
line of a normal probability plot. Only 5% of all points
will fall, by chance, outside the two curves on either side of
the line.

Clearly this data set has not been drawn from a normal distribution.

Here are some other normal probability plots (as produced by Version 19 of Minitab - similar to Versions 13 to 18).

The set-up is similar to the steps above, except for drawing the samples from other probability distributions.

In the worksheet in the data pane, |

For a sample drawn from the standard Click on the menu item " Move to " On the new pop-up menu, |

Enter 100 as the 'Number of rows of data to generate'. Click in the right pane 'Store in column(s)'. In the left pane of the dialog box, Leave the 'Mean' and 'Standard deviation' at their default values. Click 'OK'. |

Click on the menu item
" Click on
" |

As before, accept the default "Single" Just click on the " |

Minitab remembers your choices In the left pane, Then click on the 'Labels' button. |

Again Minitab remembers your most recent choice. Replace the word 'Exponential' by the word 'Normal'. Then click 'OK' on this and the previous dialog boxes. |

A new tab appears in the output pane.

100 data drawn from a standard Normal distribution N(0, 1)

Note how nearly all points lie within the two curves, close to a
straight line.

In this simulation, only one of the 100 points is clearly outside the
95% confidence bands.

The 'P-value' of 0.681 indicates that 68.1% of all future samples drawn
randomly from this population will show a greater deviation from the
ideal normal distribution than this sample does. We are
therefore very confident that this sample came from the standard
normal distribution (which, of course, it did!).

Similar steps allow us to explore normal probability plots for samples drawn at random from other probability distributions.

100 data drawn from a standard LogNormal distribution.

Note how this plot resembles that for the exponential distribution.

100 data drawn from a Cauchy distribution of mean 0 and semi-interquartile range 1.

This plot reveals the very heavy tails of the Cauchy distribution.

For the run that produced this plot, the 100 data values had a sample
median very close to the population median value of zero and quartiles
near ±1 (which usually happens).
By chance, very extreme outliers on both sides
(dozens of interquartile ranges away from the median) cancelled out to
leave this sample mean close to zero. If you draw more random
samples from the same Cauchy distribution, you will find instances when
the sample mean is far away from the population mean of zero.

Recall from the last question of Problem Set 5 that the Cauchy distribution has a finite interquartile range but an infinite population variance, which renders the sample mean completely unstable.

100 data drawn from a Beta distribution Beta(18, 2).

Note how this plot reveals a strong negative skew, together with the light right tail. The plot clearly shows that these data are inconsistent with a normal distribution.

Here is one example of a ** non**-normal probability plot.

As before, click on 'Graph' then 'Probability Plot...', then 'Single' and 'OK'.

In the 'Probability Plot: Single' dialog box,

click on the 'Distribution' button.

In the 'Probability Plot: Distribution' dialog box, first tab 'Distribution',

pull down the menu for 'Distribution' and click on 'Exponential'

Then click 'OK'. You should also adjust the label.

Now we can see that our 'Lifetime' sample is consistent with having been drawn
from an exponential distribution (which, of course, it was!)

Feel free to explore other features of Minitab’s powerful probability plot options, such as changing the confidence intervals from 95% to 99%, or changing the normal probability plot to a probability plot for another distribution.

You can export the results from the various tabs of the output pane
to Word.

Then save your work and exit from Minitab.

The populations from which these
data sets came are illustrated here, in their standard forms.

standard exponential |
standard normal |
---|---|

standard log-normal |
standard Cauchy |

beta (18, 2) |

Open Excel and import the data set, for which you wish to create a normal probability plot, into a column.

Check that the Analysis Tool Pack is present, as follows. Click on the menu item
" If it is present, then bypass the next step. Otherwise, click on
" |

Check the options for `OK` " button. |

Alongside the genuine data, The new values can be any set of They are needed for the next two steps. |

Click on the menu item
" Click on " |

Scroll down the list of Analysis Tools
and click on " then click on the " |

In the Regression dialog box: Select the range where your data are stored as the
" Select the range where the junk data are stored as the
" Choose a location for the tables of values, (most of which will be irrelevant!). In this example, a separate tab has been chosen. Check the last box " `OK` " button. |

At the location where the output was chosen to be, various tables and a graph appear.

Resize the graph and move it
to another tab.

Right-click on any of the plotted points.

Click on "`Add Trendline...`

".

In the new dialog box, accept the default option (linear trend) and

click on the "`OK`

" button.

You may also wish to tidy the graph up somewhat, to produce a result
like this:

The Excel file is available here.

Clearly Minitab does a far superior job of constructing probability plots than Excel can!

- simulate drawing a random sample from a known normal population,
- construct one- and two-sided confidence intervals for the mean,
- draw inferences on the true value of the mean from the confidence intervals.

We shall use Minitab to

In a new project, |

From the "`Calc`

" menu, select
"`Random Data`

" then click on
"`Normal...`

"

In the dialog box that pops up; enter 100 as the " select enter 1000 for the " enter 10 as the " click " |

A random sample of 100 values These values have been drawn randomly from |

From the "`Stat`

" menu, select
"`Basic Statistics`

" then click on
"`Display Descriptive Statistics...`

"

In the left pane of the dialog box, double click on ' Accept all of the defaults by just clicking
" |

The summary statistics for your random sample then appear in the
output pane.

The exact values will vary each time.

The sample mean and sample median are usually close to but not exactly 1000

and the sample standard deviation is usually close to but not exactly 10,

(and therefore the sample standard error is close to
).

Now let Minitab find a 95% two-sided confidence interval for the population
mean *µ* , based on this random sample.

From the "`Stat`

" menu, select
"`Basic Statistics`

" then click on
"`1-Sample t...`

"

In the dialog box " Click on the larger right pane. Double click on Click on the " |

In the secondary pop-up window we see and " Just accept these defaults for now and |

Using the sample mean, the sample size The result is displayed in the output pane. |

We can easily change the level of confidence and/or replace the two-sided interval by a one-sided interval.

Repeat the steps In the dialog box " Click on the " In the secondary pop-up window shown here, Click " |

The 99% one-sided confidence interval for
Speaking loosely, we are “99% sure that |

We can also conduct a classical hypothesis test, |

The additional information appears in the The The In this illustration the Of course, we know that the true population mean in |

You are encouraged to construct both 95% and 99% confidence intervals,
two-sided and one-sided, with other data sets.

Assignment #2 requires both a normal
probability plot and a one-sided confidence interval.

Also available here is a Minitab macro to simulate the construction of confidence intervals, from many random samples, for a population mean and to show that the proportion of 95% confidence intervals that capture the true value of the population mean is close to 95%.

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Created 2003 10 19 and most recently modified 2021 03 30 by Dr. G.H. George