PSY201: Chapter 9 - t-Statistic Flashcards
The t Statistic: An Alternative to z
n The t statistic allows researchers to use sample data to test hypotheses about an unknown population mean.
n The particular advantage of the t statistic, is that the t statistic does not require any knowledge of the population standard deviation.
The t Statistic: An Alternative to z
n Thus, the t statistic can be used to test hypotheses about a completely unknown population; that is, both μ and σ are unknown, and the only available information about the population comes from the sample.
The t Statistic
n All that is required for a hypothesis test with t is a sample and a reasonable hypothesis about the population mean.
n There are two general situations where this type of hypothesis test is used:
The t Statistic
The t statistic is used when a researcher wants to determine whether or not a treatment causes a change in a population mean. In this case you must know the value of μ for the original, untreated population. A sample is obtained from the population and the treatment is administered to the sample.
The t Statistic
If the resulting sample mean is significantly different from the original population mean, you can conclude that the treatment has a significant effect.
The t Statistic
Occasionally a theory or other prediction will provide a hypothesized value for an unknown population mean. A sample is then obtained from the population and the t statistic is used to compare the actual sample mean with the hypothesized population mean. A significant difference indicates that the hypothesized value for μ should be rejected.
Estimated Standard Error and the t Statistic
Whenever a sample is obtained from a population you expect to find some discrepancy or “error” between the sample mean and the population mean.
n This general phenomenon is known as sampling error.
n The goal for a hypothesis test is to evaluate the significance of the observed discrepancy between a sample mean and the population mean.
Estimated Standard Error and the t Statistic
The hypothesis test attempts to decide between the following two alternatives:
1. Is it reasonable that the discrepancy between M and μ is simply due to sampling error and not the result of a treatment effect?
Estimated Standard Error and the t Statistic
- Is the discrepancy between M and μ more than would be expected by sampling error alone? That is, is the sample mean significantly different from the population mean?
Estimated Standard Error and the t Statistic
n The critical first step for the t statistic hypothesis test is to calculate exactly how much difference between M and μ is reasonable to expect.
n However, because the population standard deviation is unknown, it is impossible to compute the standard error of M as we did with z-scores in Chapter 8.
Estimated Standard Error and the t Statistic
Therefore, the t statistic requires that you use the sample data to compute an estimated standard error of M.
Estimated Standard Error and the t Statistic
This calculation defines standard error exactly as it was defined in Chapters 7 and 8, but now we must use the sample variance, s2, in place of the unknown population variance, σ2 (or use sample standard deviation, s, in place of the unknown population
standard deviation, σ).
Estimated Standard Error and the t Statistic
The resulting formula for estimated standard error is
s2 s sM=── or sM=── n √n
Estimated Standard Error and the t Statistic
n The t statistic (like the z-score) forms a ratio.
n The numerator contains the obtained difference between the sample mean and the hypothesized population mean.
Estimated Standard Error and the t Statistic
The denominator is the standard error which measures how much difference is expected by chance.
obtained difference M - μ
t = ───────────── = ─────
standard error sM
