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Chapter 9: Introduction to the t Statistic - Coggle Diagram
Chapter 9: Introduction to the t Statistic
The estimated standard error (sM) is used as an estimate of the real standard error σM when the value of σ 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.
To maximize the similarity from one version to another, we will use variance in the formula for all of the different t statistics.
Then we can substitute the estimated standard error in the denominator of the z-score formula. The result is a new test statistic called a t statistic:
Whenever we present a t statistic, the estimated standard error will be computed as
The t statistic is used to test hypotheses about an unknown population mean, μ, when the value of σ 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.
The difference between the t formula and the z-score formula is that the z-score uses the actual population variance (or the standard deviation), and the t formula uses the corresponding sample variance (or standard deviation) when the population value isn’t known.
The greater the value of df for a sample, the better the sample variance,s2 , represents the population variance, σ2 , and the better the t statistic approximates the z-score.
How well a t distribution approximates a normal distributor is determined by degrees of freedom. In general, the greater the sample size (n) is, the larger the degrees of freedom (n - 1) are, and the better the t distribution approximates the normal distribution.
The exact shape of a t distribution changes with degrees of freedom.
The t distribution tends to be flatter and more spread out, whereas the normal z distribution has more of a central peak.
As sample size and df increase, however, the variability in the t distribution decreases, and it more closely resembles a normal distribution.
The t distribution is flatter and more spread out, especially when n is small.
A t distribution table is used to find the proportions for t statistics.
T statistics permits hypothesis testing in situations for which you do not have a known population mean to serve as a standard..
All that’s needed to compute a t statistic is a null hypothesis and a sample from the unknown population
A t test can be used in situations for which the null hypothesis is obtained from a theory, or a logical prediction
The t statistic typically requires more computation than is necessary for a z-score.
Percentage of variance accounted for by the treatment: Is a measure of effect size that determines what portion of the variability in the scores can be accounted for by the treatment effect.
Rather than computing (r2) directly by comparing two different calculations for SS, the value can be found from a single equation based on the outcome of the t test.
The letter r is the traditional symbol used for a correlation
Two basic assumptions are necessary for hypothesis tests with the t statistic.
The values in the sample must consist of independent observations.
Two observations are independent if there is no consistent, predictable relationship between the first observation and the second.
The population sampled must be normal.
This assumption is a necessary part of the mathematics underlying the development of the t statistic and the t distribution table.
The circumstances were the t statistic is used is when the population variance or standard deviation is unknown. The t statistic uses the sample variance or standard deviation in place of the unknown population values. The z-score is used however, when the population standard deviation (or variance) is known.
A confidence interval is an 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.
The construction of a confidence interval begins with the observation that every sample mean has a corresponding t value
When a t statistic is used for a hypothesis test, Cohen’s d can be computed to measure effect size. In this situation, the sample standard deviation is used in the formula to obtain an estimated value for d:
