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Introduction to the t-Statistic, image, image, image, image, image -…
Introduction to the t-Statistic
9-1 The t-Statistic: An Alternative to z
The
t statistic
is used to hypotheses about an unknown population mean, μ, when the value of s is unknown. The formula for the t statistic has the same structure as the z-score formula, except that the t statistic uses the estimated standard error in the denominator.
Degrees of freedom
describe the number of scores in a sample that are independent and free to vary. The sample mean places a restriction on the value of one score in the sample, there are
n
-1 degrees of freedom for a sample with
n
scores.
The
estimated standard error
is used as an estimate of the actual standard error, when the value of s is unknown. It is computed from the sample variance or sample standard deviation and provides an estimate of the standard distance between a sample mean
M
and the population mean μ.
9-2 Hypothesis Tests with the t-Statistic
Hypothesis Testing Example:
Step 1 -
State the Hypotheses and Select an Alpha Level
Step 2 -
Locate the Critical Region
Step 3 -
Calculate the Test Statistic
Step 4 -
Make a Decision Regarding Ho
As always, the null hypothesis states that the treatment has no effect; specifically, states that the population mean is unchanged. Thus, the null hypothesis provides a specific value for the unknown population mean. The sample data provide a value for the sample mean. Finally, the variance and estimated standard error are computed from the sample data. When these values are used in the t formula, the result becomes
t = sample mean (from the data) - population mean (hypothesized from Ho)/ estimated standard error (computed from the sample data)
Two basic assumptions are necessary for hypothesis tests with the t statistic:
The values in the sample must consist of
independent
observations
The population sampled must be normal
9-1 The t-Statistic: An Alternative to z
A
t distribution
is the complete set of t values computed for every possible random sample for a specific sample size (
n
) or a specific degrees of freedom (
df
). The t distribution approximates the shape of a normal distribution. As df gets very large, the t distribution gets closer in shape to a normal z-score distribution. distributions of t are bell-shaped and symmetrical and have a mean of zero. However, the t distribution has more variability than a normal z distribution, especially when df values are small, the t distribution tends to be flatter and more spread out, whereas the normal z distribution has more of a central peak
9-3 Measuring Effect Size for the t-Statistic
Cohen's
d
= mean difference/standard deviation
estimated
d
= mean difference/sample standard deviation
A
confidence interval
, or range of values centered around a sample statistic. The logic behind a confidence interval is that a sample statistic, such as a sample mean, should be relatively near to the corresponding population parameter. Therefore, we can confidently estimate that the value of the parameter should be located in the interval near to the statistic.
9-4 Directional Hypotheses and One-Tailed Test
The steps are the same as making a directional hypotheses and one tailed tests:
Step 1 -
State the Hypotheses and Select an Alpha Level.
Step 2 -
Locate the Critical Region.
Step 3 -
Calculate the Test Statistic.
Step 4 -
Make a Decision
