problem with z-scores
z-score formula requires more information than is usually available. Specifically, a z-score requires that we know the value of the population standard deviation (or variance), which is needed to compute the standard error.
equation for estimated standard error
standard error (sm) = s / square root of n
estimated standard error
The estimated standard error is used as an estimate of the real standard error when the value of the standard deviation 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.
There are two reasons for making this shift from standard deviation to variance:
- the sample variance is an unbiased statistic; on average, the sample variance provides an accurate and unbiased estimate of the population variance . Therefore, the most accurate way to estimate the standard error is to use the sample variance to estimate the population variance.
- in the formulas for estimated standard error. To maximize the similarity from one version to another, we will use variance in the formula for all of the different t statistics.
whenever we present a t statistic, the estimated standard error will be computed as estimated standard error = square root of (sample variance/sample size)
t-statistic equation
t = M-mu / sm (mean - population mean / estimated standard error)
t statistic
The t statistic is used to test hypotheses about an unknown population mean, μ, when the value of the standard deviation 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.
degrees of freedom
describe the number of scores in a sample that are independent and free to vary. Because the sample mean places a restriction on the value of one score in the sample, there are n − 1 degrees of freedom for a sample with n scores
- The greater the value of df for a sample, the better the sample variance, , represents the population variance, , and the better the t statistic approximates the z-score.
a t distribution approximates a normal distributor
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 are, and the better the t distribution approximates the normal distribution.
t distribution
The distribution of t statistics is symmetrical and centered at zero like a normal distribution. A t distribution is flatter and more spread out than the normal distribution, but approaches a normal shape as df increases.
shape of the t distribution
- The exact shape of a t distribution changes with degrees of freedom.
- For t statistics, the bottom of the formula varies from one sample to another. Specifically, the sample variance changes from one sample to the next, so the estimated standard error also varies. Thus, 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. As sample size and df increase, however, the variability in the t distribution decreases, and it more closely resembles a normal distribution.
When these values are used in the t formula, the result becomes
t = sample mean (from the data) - population mean (hypothesised from the null hypothesis) / estimated standard error (computed from the sample data)
Hypothesis testing steps
- state the hypothesis and select the alpha level
- locate the critical region
- calculate the test statistic
- make a decision regarding the null hypothesis
assumptions of the t-test
- The values in the sample must consist of independent observations.
- The population sample must be normal.
Cohen’s d or estimated d
estimated d = mean difference / sample standard deviation = M-mu / s
proportion or percentage of the total variability
variability accounted for / total variability
percentage of variance accounted for by the treatment (r^2)
A measure of effect size that determines what portion of the variability in the scores can be accounted for by the treatment effect.
r^2 = t^2 / t^2 + df
- The letter r is the traditional symbol used for a correlation
confidence intervals
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.
equation to find mean population
M +- t*sm
Two characteristics of the confidence interval should be noted.
- 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.
- 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).
