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Distributions Menu
The Distributions menu
selection is used to calculate critical
values and probabilities for various
distributions. The most common
distributions are the z (normal)
distribution, t distribution, F
distribution, and the chi-square
distribution. Within the last 20 years,
computers have made it easy to calculate
exact probabilities for the various
statistics. Prior to that, researchers
made extensive use of books containing
probability tables.
The Distributions menu has four
selections:
Normal distribution
The normal distribution is the most
well-known distribution and is often
referred to as the z distribution
or the bell shaped curve. It is used when
the sample size is greater than 30. When
the sample size is less than 30, the t
distribution is used instead of the
normal distribution.
The menu offers three choices: 1)
probability of a z value, 2) critical z
for a given probability, and 3)
probability of a defined range.
Probability of a z value
When you have a z (standardized) value
for a variable, you can determine the
probability of that value. The software
is the electronic equivalent of a normal
distribution probability table. When you
enter a z value, the area under
the normal curve will be calculated. The
area not under the curve is
referred to as the rejection region.
It is also called a two-tailed
probability because both tails of the
distribution are excluded. The Statistics
Calculator reports the two-tailed
probability for the z value. A one-tailed
probability is used when your research
question is concerned with only half of
the distribution. Its value is exactly
half the two-tailed probability.
Example
z-value: 1.96
-----------------------------------------
Two-tailed
probability = .05
Critical z for a given probability
This menu selection is used to
determine the critical z value for
a given probability.
Example
A large company
designed a pre-employment survey to be
administered to prospective employees.
Baseline data was established by
administering the survey to all current
employees. They now want to use the
instrument to identify job applicants who
have very high or very low scores.
Management has decided they want to
identify people who score in the upper
and lower 3% when compared to the norm.
How many standard deviations away from
the mean is required to define the upper
and lower 3% of the scores?
The total area
of rejection is 6%. This includes 3% who
scored very high and 3% who scored very
low. Thus, the two-tailed probability is
.06. The z value required to reject 6% of
the area under the curve is 1.881. Thus,
new applicants who score higher or lower
than 1.881 standard deviations away from
the mean are the people to be identified.
Two tailed
probability: .06
---------------------------------
z-value = 1.881
Probability of a defined range
Knowing the mean and standard
deviation of a sample allows you to
establish the area under the curve for
any given range. This menu selection will
calculate the probability that the mean
of a new sample would fall between two
specified values (i.e., between the
limits of a defined range).
Example
A manufacturer
may find that the emission level from a
device is 25.9 units with a standard
deviation of 2.7. The law limits the
maximum emission level to 28.0 units. The
manufacturer may want to know what
percent of the new devices coming off the
assembly line will need to be rejected
because they exceed the legal limit.
Sample mean =
25.9
Unbiased standard deviation = 2.7
Lower limit of the range = 0
Upper limit of the range = 28.0
----------------------------------------------------------------
Probability of a
value falling within the range = .7817
Probability of a value falling outside
the range = .2183
The area under
the curve is the sum of the area defined
by the lower limit plus the area defined
by the upper limit.
The area under
the normal curve is the probability that
additional samples would fall between the
lower and upper limits. In this case, the
area above the upper limit is the
rejection area (21.83% of the product
would be rejected).
T distribution
Mathematicians used to think that all
distributions followed the bell shaped
curve. In the early 1900's, an Irish
chemist named Gosset, discovered that
distributions were much flatter than the
bell shaped curve when working with small
sample sizes. In fact, the smaller the
sample, the flatter the distribution. The
t distribution is used instead of
the normal distribution when the sample
size is small. As the sample size
approaches thirty, the t
distribution approximates the normal
distribution. Thus, the t distribution is
generally used instead of the z
distribution, because it is correct for
both large and small sample sizes, where
the z distribution is only correct for
large samples.
The menu offers three choices: 1)
probability of a t value, 2) critical t
value for a given probability, and 3)
probability of a defined range.
Probability of a t-value
If you have a t value and the
degrees of freedom associated with the
value, you can use this program to
calculate the two-tailed probability of t.
It is the equivalent of computerized
table of t values.
Example
t-value: 2.228
df: 10
------------------------------------
Two-tailed
probability = .050
Critical t value for a given probability
This program is the opposite of the
previous program. It is used if you want
to know what critical t value is
required to achieve a given probability.
Example
Two-tailed
probability: .050
Degrees of freedom: 10
-----------------------------------
t-value = 2.228
Probability of a defined range
Knowing the mean and standard
deviation of a sample allows you to
establish the area under the curve for
any given range. You can use this program
to calculate the probability that the
mean of a new sample would fall between
two values.
Example
A company did a
survey of 20 people who used its product.
The mean average age of the sample was
22.4 years and the unbiased standard
deviation was 3.1 years. The company now
wants to advertise in a magazine that has
a primary readership of people who are
between 18 and 24, so they need to know
what percent of its potential customers
are between 18 and 24 years of age?
Sample mean:
22.4
Unbiased standard deviation: 3.1
Sample size = 20
Lower limit of the range = 18
Upper limit of the range = 24
----------------------------------------------------------------
Probability of a
value falling within the range = .608
Probability of a value falling outside
the range = .392
Because of the
small sample size, the t distribution is
used instead of the z distribution. The
area under the curve represents the
proportion of customers in the population
expected to be between 18 and 24 years of
age. In this example, we would predict
that 60.8% of the its customers would be
expected to be between 18 and 24 years of
age, and 39.2% would be outside of the
range. The company decided not to
advertise.
F distribution
The F-ratio is used to compare
variances of two or more samples or
populations. Since it is a ratio (i.e., a
fraction), there are degrees of freedom
for the numerator and denominator. This
menu selection may be use to calculate
the probability of an F -ratio or
to determine the critical value of F
for a given probability. These menu
selections are the computer equivalent of
an F table.
Probability of a F-ratio
If you have a F-ratio and the
degrees of freedom associated with the
numerator and denominator, you can use
this program to calculate the
probability.
Example
F-ratio: 2.774
Numerator degrees of freedom: 20
Denominator degrees of freedom: 10
----------------------------------------------
Two-tailed
probability = .0500
Critical F for a given probability
If you know the critical alpha level
and the degrees of freedom associated
with the numerator and denominator, you
can use this program to calculate the
F-ratio.
Example
Two-tailed
probability = .0500
Numerator degrees of freedom: 20
Denominator degrees of freedom: 10
-----------------------------------------------
F-ratio: 2.774
Chi-square distribution
The chi-square statistic is used to
compare the observed frequencies in a
table to the expected frequencies. This
menu selection may be use to calculate
the probability of a chi-square statistic
or to determine the critical value of
chi-square for a given probability. This
menu selection is the computer equivalent
of an chi-square table.
Probability of a chi-square statistic
If you have a chi-square value and the
degrees of freedom associated with the
value, you can use this program to
calculate the probability of the
chi-square statistic. It is the
equivalent of computerized table of
chi-square values.
Example
Chi-square
value: 18.307
Degrees of freedom: 10
------------------------------------
Probability =
.050
Critical chi-square for a given probability
If you have the critical alpha level
and the degrees of freedom, you can use
this program to calculate the probability
of the chi-square statistic. It is the
equivalent of computerized table of
chi-square values.
Example
Probability =
.0500
Degrees of freedom: 10
------------------------------------
Chi-square
value: 18.307
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