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The t Statistic: An Alternative to z - Coggle Diagram
The t Statistic: An Alternative to z
T
he Problem with z-Scores
: z-score requires that we know the value of the population standard deviation or the variance, which is needed to compute the standard error.
The
estimated standard error of M**
: is sm which indicates that the estimated value is computed from sample data rather than from the actual population parameter.
The
t-statistic
is a value used in statistics to determine how much a sample mean differs from a hypothesized population mean in units of standard error. It’s commonly used in t-tests when the sample size is small or the population standard deviation is unknown.:
The
t-statistic** measures how many standard errors the sample mean is away from the hypothesized mean.
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:
Degrees of freedom (df) is a concept in statistics that refers to the number of independent pieces of information in a calculation that are free to vary. Degrees of freedom are important because they affect the shape of the t-distribution:
The t- Distribution
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
The Shape of the t Distribution
changes with degrees of freedom.
Determining Proportions and Probabilities for t Distributions
: involves finding the area under the t-curve corresponding to a given t-value, taking into account the degrees of freedom (df).
Hypothesis Tests with the t Statistic
Hypothesis tests with the t-statistic are used when we want to compare a sample mean to a population mean (or compare two sample means and the population standard deviation is unknown, which is common in real-world data.:
The
t statistic
is used when the population variance (or standard deviation) is unknown.
The Unknown Population
exists after the treatment is administered, and the null hypothesis simply states that the value of the mean is not changed by the treatment.
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