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Chapter 9 - Introduction to the t-Statistic - Coggle Diagram
Chapter 9 - Introduction to the t-Statistic
9-1 The
t
Statistic: An Alternative to z
The Problem with z-Scores
the z-score formula requires more info than is usually available
requires that we know the value of the population SD (or variance) - needed to compute standard error
When the variance (or SD) is not known, we use the corresponding sample value in its place.
Introducing the
t
Statistic
standard error
estimated standard error
- an estimate of the actual standard error, when the value of s in unknown
t
statistic
- used to test hypotheses about an unknown μ, when the value of s is unknown
only difference between the
t
formula and the z-score formula is: the z-score uses the actual population variance (or the SD), and the
t
formula uses the corresponding sample variance (or SD) when the population value is not known
Degrees of Freedom and the
t
Statistic
As sample size increases, so does the value of for degrees of freedom, and sample variance will be a better estimate of population variance
describes how well
t
estimates z
The
t
Distribution
the complete set of
t
values computed for every possible random sample for a specific sample size or a specific degrees of freedom; approximates the shape of a normal distribution
t
distributions are bell-shaped and symmetrical and have a mean of zero
t
distributions are more variable than the normal distribution as indicated by the flatter and more spread-out shape
the larger the value of df is, the more closely the
t
distribution approximates a normal distribution
The Shape of the
t
Distribution
exact shape of
t
distribution changes with df
As df gets very large, the
t
distribution gets closer in shape to a normal z-score distribution
tends to be flatter and more spread out, whereas the normal z distribution has more of a central peak
t
statistics are more variable than are z-scores
as sample size and df increase, the variability in the
t
distribution decreases, and it more closely resembles a normal distribution
REVIEW: Statistical Procedures permitting the use of a sample mean to test hypotheses about an unknown population mean:
Sample mean
M
is expected to approximate its population mean (μ)
Standard error provides a measure of how much difference is reasonable to expect between a
M
and μ
To test the hypothesis, we compare the obtained
M
with the hypothesized μ by computing a z-score test statistic.
9-2 Hypothesis Tests with the
t
Statistic
Using the
t
Statistic for Hypothesis Testing
permits hypothesis testing in situations for which you do not have a known population mean to serve as a standard
all needed to compute a t statistic is a null hypothesis and a sample from the unknown population
Assumptions of the
t
Test
Values in the sample must consist of independent observations.
The population sampled must be normal
The Influence of Sample Size and Sample Variance
number of scores in the sample and magnitude of the sample variance both have a larger effect on the t statistic - influencing the statistical decision
2 factors that determine the size of the standard error are: sample variance and sample size
estimated standard error is directly related to the sample variance so that the larger the variance, the larger the error.
9-3 Measuring Effect Size for the
t
Statistic
Estimated Cohen's
d
Measuring the Percentage of Variance Explained
results when removing the treatment effect to reduce variability
Confidence Intervals for Estimating μ
estimating an unknown population mean involves constructing a confidence interval
based on the observation that a sample mean tends to provide a reasonably accurate estimate of the population mean
an interval, or range of values centered around a sample statistic
can confidently estimate that the value of the parameter should be located in the interval near to the statistic
Constructing a Confidence Interval
Factors Affecting the Width of a Confidence Interval
there is a trade-off between precision (width of interval) and confidence that interval contains population mean
the bigger the sample, the samller the interval
9-4 Directional Hypotheses and One-Tailed Tests
The Critical Region for a One-Tailed Test
determine whether the sample mean is in the direction predicted by the original research question
determine whether the effect is large enough to be significant
nondirectional = 2 tailed (more commonly used)
directional (one-tailed)
may be used in some situations: exploratory investigations (pilot studies), or when there is a priori justification
