Please enable JavaScript.
Coggle requires JavaScript to display documents.
Introduction to the t statistic - Coggle Diagram
Introduction to the t statistic
The t Statistic: An Alternative to z
Degrees of Freedom and the t Statistic
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 degrees of freedom for a sample with n scores.
degrees of freedom = df = n - 1
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.
Introducing the t Statistic
estimated standard error: used as an estimate of the actual standard error when the value of s 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
t statistic: used to test hypotheses about an unknown population mean when the value of s 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.
Determining Proportions and Probabilities for t Distributions
Just as we used the unit normal table to locate proportions associated with z-scores, we use a t distribution table to find proportions for t statistics
As df increases, the t distribution becomes more similar to a normal distribution, ultimately reaching ±1.96, the z-score values that separate the extreme 5% in a normal distribution
The Problem with z-Scores
The shortcoming of using a z-score for hypothesis testing is that the 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.
The Shape of the t Distribution
The exact shape of a t distribution changes with degrees of freedom. As df gets very large, the t distribution gets closer in shape to a normal z-score distribution.”
For t statistics, the sample variance changes from one sample to the next, so the estimated standard error also varies. As a result, t statistics are more variable than z-scores, and the distribution is flatter and more spread out.
Hypothesis Tests with the t Statistic
Hypothesis Testing Example
Step 2: Locate the Critical Region
Step 3: Calculate the Test Statistic.
Step 1: State the Hypotheses and Select an Alpha Level.
Step 4: Make a Decision Regarding H0
Assumptions of the t Test
The values in the sample must consist of independent observations.
The population sampled must be normal.
Using the t Statistic for Hypothesis Testing
The t test does not require any prior knowledge about the population mean or the population variance… all you need to compute a t statistic is a null hypothesis and a sample from the unknown population.
t = sample mean (from the data) - population mean (hypothesized from H0)/estimated standard error (computed from the sample data)
When the obtained difference between the data and the hypothesis is much greater than expected, we obtain a large value for t, concluding that the data is not consistent with the hypothesis, and our decision is to ‘reject H0'
The Influence of Sample Size and Sample Variance
The estimated standard error is inversely related to the number of scores in the sample. Large samples tend to produce bigger t statistics and therefore are more likely to produce significant results.
The estimated standard error is directly related to the sample variance so that the larger the variance, the larger the error. Large variance means that you are less likely to obtain a significant treatment effect.
Measuring Effect Size for the t Statistic
Confidence Intervals for Estimating μ
Confidence Interval: 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. Therefore, we can confidently estimate that the value of the parameter should be located in the interval near to the statistic.
Constructing a Confidence Interval
The construction of a confidence interval begins with the observation that every sample mean has a corresponding t value defined by the equation:
t = M - (μ)/SM
Measuring the Percentage of Variance Explained, r^2
An alternative method for measuring effect size is to determine how much of the variability in the scores is explained by the treatment effect. Total variability minus variability not explained by the treatment effect is equal to the amount of variability accounted for by the treatment.
Although sample size affects the hypothesis test, this factor has little or no effect on measures of effect size. High variance reduces the likelihood of rejecting the null hypothesis and it reduces measures of effect size.
Factors Affecting the Width of a Confidence Interval
A larger level of confidence produces a larger t value and a wider interval. There is a trade-off between precision (the width of the interval) and the confidence one has that the interval contains the population mean
The bigger the sample (n), the smaller the interval. A bigger sample gives you more information about the population and allows you to make a more precise estimate (a narrower interval)
Estimated Cohen's d
Cohen defined this measure of effect size in terms of the population mean difference and the population standard deviation. However, in most situations the population values are not known and you must substitute the corresponding sample values in their place.
The sample standard deviation in the denominator standardizes the mean difference into standard deviation units. An estimated d of 1.00 indicates that the size of the treatment effect is equivalent to one standard deviation
Directional Hypotheses and One-Tailed Tests
The Critical Region for a One-Tailed Test
The first stage is simply to determine whether the sample mean is in the direction predicted by the original research question. The sign of the t statistic (+ or −) no longer matters.
The second stage is to determine whether the effect is large enough to be significant. If the magnitude of the t statistic is greater than 1.860, the result is significant and H0 is rejected.
State the Hypotheses, and Select an Alpha Level.
State the Hypotheses, and Select an Alpha Level.
Calculate the Test Statistic.
Make a Decision