Week 7 Chapter 9 Flashcards
SS/df (first step in Calculation of the t statistic)
calculation of the sample variance as a substitute for the unknown population value
√(s^2 aka sample variance/n) (second step in Calculation of the t statistic)
estimation of the standard error
(M - μ)/sM (note: M is a subscript. sM is the standard error) (third step in Calculation of t statistic)
computation of the t statistic using the estimated standard error
estimated standard error of M
all of these:
1. approximation of the standard distance between a sample mean and the population mean
2. sM 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.
t statistic
hypothesis testing tool in which estimated standard error is used in the z-score formula denominator
degree of freedom
figure in a sample that is independent and can vary
t distribution
all of these:
- complete set of t values computed for every possible random sample for a sample size
- 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.
estimated Cohen’s d
figure calculated when substituting sample values in place of population values
percentage of variance accounted for by the treatment
measurement of reduction in variability after removing the treatment effect
confidence interval
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.
The shortcoming of using a z-score for hypothesis testing is that the z-score formula requires
more information than is usually available
a z-score requires that we know the value of the population standard deviation (or variance), which is needed to compute the standard error. In most situations, however, the standard deviation for the population is
not known
Like the 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 exact shape of a t distribution changes with
degrees of freedom
As df gets very large, the t distribution gets
loser in shape to a normal z-score distribution.
The t distribution tends to be flatter and more spread out, whereas the normal z distribution has
more of a central peak.
only the numerator of the z-score formula varies, but both the numerator and the denominator of the t statistic vary. As a result, t statistics are more
variable than are z-scores, and the t distribution is flatter and more spread out.
In what circumstances is the t statistic used instead of a z-score for a hypothesis test?
The t statistic is used when the population variance (or standard deviation) is unknown.
On average, what value is expected for the t statistic when the null hypothesis is true?
0
The numerator measures the actual difference between the sample data (M) and the population hypothesis (μ)
The estimated standard error in the denominator measures how much difference is reasonable to expect between a sample mean and the population mean.
When the obtained difference between the data and the hypothesis (numerator) is much greater than expected (denominator), we obtain a large value for t (either large positive or large negative). In this case, we conclude that
the data are not consistent with the hypothesis, and our decision is to “reject H0.” On the other hand, when the difference between the data and the hypothesis is small relative to the standard error, we obtain a t statistic near zero, and our decision is “fail to reject H0.”
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.
The population sampled must be normal assumption of t statistic
is a necessary part of the mathematics underlying the development of the t statistic and the t distribution table. However, violating this assumption has little practical effect on the results obtained for a t statistic, especially when the sample size is relatively large. With very small samples, a normal population distribution is important. With larger samples, this assumption can be violated without affecting the validity of the hypothesis test. If you have reason to suspect that the population distribution is not normal, use a large sample to be safe.
A sample is selected from a population and a treatment is administered to the sample. For a hypothesis test with a t statistic, if there is a 5-point difference between the sample mean and the original population mean, which set of sample characteristics is most likely to lead to a decision that there is a significant treatment effect?
Small variance for a large sample
A researcher selects a sample of n = 12 scores from a normal population with a mean of μ = 100. A treatment is administered to the individuals in the sample. After treatment, the sample mean is M = 108. Based on this result, is it reasonable to conclude that the treatment has an effect?
The answer depends on the variance (or standard deviation) for the sample.
If other factors are held constant, how does the size of the sample variability (either the standard deviation or variance) affect a hypothesis test?
Larger variability, as defined by a larger standard deviation, tends to decrease the value of the t statistic (moves t closer to zero) and reduces the likelihood of rejecting the null hypothesis.
To gain more confidence in your estimate, you must
increase the width of the interval. Conversely, to have a smaller, more precise interval, you must give up confidence.
In the estimation formula, the percentage of confidence influences the value of
t
A larger level of confidence (the percentage), produces
a larger t value and a wider interval.
the bigger the sample (n), the smaller the interval. This relationship is straightforward if you consider the sample size as a measure of the amount of information. A bigger sample gives you more information about the population and allows you to make a more
precise estimate (a narrower interval). The sample size controls the magnitude of the standard error in the estimation formula. As the sample size increases, the standard error decreases, and the interval gets smaller.
if a researcher feels that an interval is too broad (producing an imprecise estimate of the mean), the interval can be narrowed by
either increasing the sample size or lowering the level of confidence.
confidence intervals are influenced by sample size, they do not provide an unqualified measure of absolute effect size and are
not an adequate substitute for Cohen’s d or r^2 (r squared). Nonetheless, they can be used in a research report to provide a description of the size of the treatment effect.
occasionally a t value is so extreme that the computer reports = 0.000. The zero value does not mean that the probability is literally zero; instead, it means that the computer has rounded off the probability value to three decimal places and obtained a result of 0.000. In this situation, you do not know the exact probability value, but you can report
p < 0.001
A researcher fails to reject the null hypothesis with a regular two-tailed test using a = .05. If the researcher used a directional (one-tailed) test with the same data and the same alpha level, then what decision would be made?
Possibly reject the null hypothesis if the treatment effect is in the predicted direction
