To compare the mean value of the population with a certain value or with the mean value of other populations can be done with the z test . However, the z test can only be used if the data is normally distributed and the population variance is known .
In fact, it is rare that we can know the parameter values of a population with certainty, so we can only estimate the parameters of the population from the sample we take. Since we do not know what the population standard deviation, this value is estimated by the sample standard deviation, s, which is calculated from the sample. However, for small samples, s is not an accurate estimate for so it is no longer valid if we use it for the z test. For small sample sizes, we can approach it by using the student's t-test .
A full discussion of the t-student test can be read in the following document.
Characteristics of the t-distribution
- The shape of the t-distribution is similar to the normal distribution, it is bell-shaped and symmetrical with the value of t = 0 at its midpoint.
- The t distribution has a wider variance than the normal distribution. The variance value > 1 while the normal distribution value = 1.
- It has a degree of freedom (n-1), where n is the sample size.
- As the sample size gets larger, the shape of the t-distribution is almost close to the Normal distribution. This is because with the larger the sample size, the variance value will be close to 1.
In statistics, there are four types of statistical t-test, namely:
- Test the hypothesis that the mean of the population is equal to a certain value
- Test the hypothesis for the difference of two means of random samples of the equal variance
- Test the hypothesis for the difference of two means of random samples with unequal variance
- Test the hypothesis for the mean of the paired samples
Steps in Hypothesis Testing
- Establish the formulation of the hypothesis to be tested, whether one tailed test or two-tailed test. The hypothesis (claim) is expressed in the form of a symbol and also gives a symbolic form for statements outside the hypothesis (if the conjecture is false) for example:
- If the hypothesis has changed, then the symbol: μ ≠ D0 (beyond that μ = D0),
- If the hypothesis: the new method is better, then the symbol: μ > D0 (beyond that μ = D0).
- If the hypothesis is: least (minimal), then the symbol: μ ≥ D0 (beyond that μ < D0).
- Of the two symbolic statements, specify H1 for statements that do not contain equations (>, <, ≠) and H0 for statements containing equation marks (at point 1, the symbol statement for H0 is bolded).
- Determine the desired significant level (α).
- Determine the appropriate type of statistical test based on known data and information both from the population and from samples taken from that population.
- Calculate the t-critical value, draw a sketch to make it easier to find the territory of rejection and acceptance
- Reject H0 when the t-stat value falls in the critical region, which is located in the H0 rejection region and vice versa, accept H0 when the test statistical value is located in the H0 acceptance area.
Assumption
To use the t-test, it must meet the following requirements (assumptions):
- The population should spread normally or n > 30
- Unknown population variance
- Observations are independent and randomly drawn from the population
The assumption of the population spreading normally is not so necessary if:
- Number of samples > 30
- The histogram almost resembles a precarious or not too far from the normal distribution form and there is no outlier
Calculation
Almost all statistical software uses modern methods of testing a hypothesis, namely by using the probability value (p-value) to state whether the hypothesis is significant or not (usually symbolized by Sig. or p-value). In contrast to traditional calculation methods that compare statistical test values (e.g. tstat) with critical values (e.g. t-tables).
Differences between modern methods and traditional methods in the case of t-tests:
Traditional Methods | Modern Methods | |
Significant (H0 rejected) | |value tstat| > | value ttable.| example: | -2,724| > | 2,056| | its opportunity value (Sig.) < the specified significant level e.g.: Sig = 0.011 < 0.05 |
Not significant (H0 accepted) | |value tstat| < | value ttable.|
| its opportunity value (Sig.) > the significant extent specified |
