```
Labortory Exercise 2: Descriptive Statistics
--------------------------------------------

Density Curve, Normal Quantile Plot

An accurate measure of the reaction time is of great importance for
the interpretation of experiments involving reactions of
subjects. Below 64 reaction times.

28  22  36  26  28  28  26  24  32  30  27
24  33  21  36  32  31  25  24  25  28  36
27  32  34  30  25  26  26  25  23  21  30
33  29  27  29  28  22  26  27  16  31  29
36  32  28  40  19  37  23  32  29  24  25
27  24  16  29  20  28  27  39  23

a. A histogram can show the features of a distribution that are
obviously not normal, like peaks, clear skewness or gaps and clusters.
Draw a histogram for the given times and combine them with a normal
curve. Do you think the distribution is normal? What is the meaning of
the total area under the curve? What is this area equal to?

b. Calculate mean and standard deviation.

c. A standardized observation z shows the distance of an original
observation x to the mean in numbers of standard deviations,
including the direction (positive means above, negative below the
mean).  Assuming that the distribution is perfectly normal,
validates the following formula:

X - muu
z = -------
sigma

Convert the x-scores to the standardized z-score. What is the
z-score for x=37?

d. Create a scatterplot from the data. Put the x-scores on the x-axis
and the z-scores on the y-axis.

e. Sort the tabel so that the x-scores are arranged in size, in
ascending order.  Next, add a new column and put in position
numbers from 1 to 64. Which position is assigned to x=37?

f. Calculate all quantiles by dividing all sortingnumbers by the total
amount of observations (64). Which quantile is assigned to x=37?

g. A quantile corresponding with a z-score (or nonstandardized: an
x-score) gives the relative frequency of all values smaller or
equal to z (or: x). Or: the chance getting a value smaller or equal
to z (or: x). Described by formulas: P(Z<=z) (or: P(X<=x)).
Look for the z-score of the quantile (or the P(Z<=z)) of x=37 in
table A, and give that score.

h. In a normal-quantile-plot, each x-score is set out against the
z-score.  The z-score is derived from the quantile that corresponds
with the x-value (using table A).  In the graph, the x-scores are
put on the x-axis, the z-scores on the y-axis. Draw the
normal-quantile-plot. Combine this with the line z=x (this is the
line connecting the dots of the scatterplot in d.). Is it a normal
distribution?

```