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 normal probability plot for a
case where the data are known to be non-normal, namely data drawn
from an exponential distribution. The standard exponential
distribution Start Minitab. Click on the menu item " Move to " On the new pop-up menu, click on
" |

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
follow an exponential distribution, we shall use
the name The column Note that, as this is a set of random data, the numbers in your data column will not be identical to those shown here. |

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 dialog box, Double click on " Click on the " |

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

Two graph windows then appear:

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 Session window,

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 dialog box,

Double click on "`C1 Lifetime`

"
to select it into the "`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:

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 17 of Minitab - mostly the same as Versions 13 to 16 and 18).

100 data drawn from a Normal
distribution N(1000, 10^{2})

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

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. However, the sample mean was well above +7.
Recall from the last question of Problem Set 5
that the Cauchy
distribution has a finite interquartile range but an infinite
population variance.

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.

You can copy and paste the above results into your Report Pad.

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:

- 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; |

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...`

"

Accept the defaults by just clicking "`OK`

" on
the dialog box that pops up.

The summary statistics for your random sample then appear in the
Session window.

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 " Just accept these defaults for now and |

Using the sample mean, the sample size |

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 " 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 In this illustration the Of course, we know that the true population mean in this case is actually 1000 exactly. |

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 2018 08 21 by Dr. G.H. George