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Although
there are numerous sampling distributions used in hypothesis testing, the normal
distribution is the most common example of how data would appear if we created a frequency
histogram where the x axis represents the values of scores in a distribution and the y
axis represents the frequency of scores for each value.
Most scores will be similar and therefore will group near the center of the
distribution. Some scores will have unusual
values and will be located far from the center or apex of the distribution. These unusual scores are represented below as the
shaded areas of the distribution. In
hypothesis testing, we must decide whether the unusual values are simply different because
of random sampling error or they are in the extreme tails of the distribution because they
are truly different from others. Sampling
distributions have been developed that tell us exactly what the probability of this
sampling error is in a random sample obtained from a population that is normally
distributed.
Properties
of a normal distribution
·
Forms
a symmetric bell-shaped curve
·
50%
of the scores lie above and 50% below the midpoint of the distribution
·
Curve
is asymptotic to the x axis
·
Mean,
median, and mode are located at the midpoint of the x axis
Using
theoretical sampling probability distributions
Sampling
distributions allow us to approximate the probability that a particular value would occur
by chance alone. If you collected means from
an infinite number of repeated random samples of the same sample size from the same
population you would find that most means will be very similar in value, in other words,
they will group around the true population mean. Most
means will collect about a central value or midpoint of a sampling distribution. The frequency of means will decrease as one travels
away from the center of a normal sampling distribution.
In a normal probability distribution, about 95% of the means resulting from
an infinite number of repeated random samples will fall between 1.96 standard errors above
and below the midpoint of the distribution which represents the true population mean and
only 5% will fall beyond (2.5% in each tail of the distribution).
The
following are commonly used points on a distribution for deciding statistical
significance.
90%
of scores
+/- 1.65
standard errors
95%
of scores
+/- 1.96
standard errors
99%
of scores
+/- 2.58
standard errors
Standard
error: Mathematical adjust to the standard deviation to
account for the effect sample size has on the underlying sampling distribution. It represents the standard deviation of the
sampling distribution.
Alpha
and the role of the distribution tails
The
percentage of scores beyond a particular point along the x axis of a sampling distribution
represent the percent of the time during an infinite number of repeated samples one would
expect to have a score at or beyond that value on the x axis. This value on the x axis is known as the critical
value when used in hypothesis testing. The midpoint represents the actual population
value. Most scores will fall near the actual population value but will exhibit some
variation due to sampling error. If a score
from a random sample falls 1.96 standard errors or farther above or below the mean of the
sampling distribution, we know from the probability distribution that there is only a 5%
or less chance of randomly selecting a set of scores that would produce a sample mean that
far from the true population mean. This area
above and below 1.96 standard errors is the region of rejection.
When
conducting significance testing, if we have a test statistic that is at least 1.96
standard errors above or below the mean of the sampling distribution, we assume we have a
statistically significant difference between our sample mean and the expected mean for the
population. Since we know a value that far
from the population mean will only occur randomly 5% or less of the time, we assume the
difference is the result of a true difference between the sample and the population mean,
and is not the result of random sampling error. The
5% is also known as the probability of being wrong when we conclude statistical
significance.
1-tailed
vs. 2-tailed statistical tests
A
2-tailed test is used when you cannot determine a priori whether a difference between
population parameters will be positive or negative. A
1-tailed test is used when you can reasonably expect a difference will be positive or
negative. If you retain the same critical
value for a 1-tailed test that would be used if a 2-tailed test was employed, the alpha
is halved (i.e., .05 alpha would become .025 alpha).
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