Null and Alternative Hypotheses
In a hypothesis test, sample data is evaluated in order to arrive at a decision about some type of claim. If certain conditions about the sample are satisfied, then the claim can be evaluated for a population. In a hypothesis test, we:
- Evaluate the null hypothesis, typically denoted with \(H_0\). The null is not rejected unless the hypothesis test shows otherwise. The null statement must always contain some form of equality (=, \(\leq\) or \(\geq\) )
- Always write the alternative hypothesis, typically denoted with \(H_a\) or \(H_1\), using not equal, less than or greater than symbols, i.e., ( \(\neq,<\), or > ).
- If we reject the null hypothesis, then we can assume there is enough evidence to support the alternative hypothesis.
- Never state that a claim is proven true or false. Keep in mind the underlying fact that hypothesis testing is based on probability laws; therefore, we can talk only in terms of non-absolute certainties.
Outcomes and the Type I and Type II Errors
In every hypothesis test, the outcomes are dependent on a correct interpretation of the data. Incorrect calculations or misunderstood summary statistics can yield errors that affect the results. A Type I error occurs when a true null hypothesis is rejected. A Type II error occurs when a false null hypothesis is not rejected.
The probabilities of these errors are denoted by the Greek letters \(\alpha\) and \(\beta\), for a Type I and a Type II error respectively. The power of the test, \(1-\beta\), quantifies the likelihood that a test will yield the correct result of a true alternative hypothesis being accepted. A high power is desirable.
Probability Distribution Needed for Hypothesis Testing
In order for a hypothesis test's results to be generalized to a population, certain requirements must be satisfied.
When testing for a single population mean:
- A Student's \(t\)-test should be used if the data come from a simple, random sample and the population is approximately normally distributed, or the sample size is large, with an unknown standard deviation.
- The normal test will work if the data come from a simple, random sample and the population is approximately normally distributed, or the sample size is large.
When testing a single population proportion use a normal test for a single population proportion if the data comes from a simple, random sample, fill the requirements for a binomial distribution, and the mean number of success and the mean number of failures satisfy the conditions: \(n p>5\) and \(n q>5\) where \(n\) is the sample size, \(p\) is the probability of a success, and \(q\) is the probability of a failure.
Full Hypothesis Test Examples
The hypothesis test itself has an established process. This can be summarized as follows:
- Determine \(H_0\) and \(H_a\). Remember, they are contradictory.
- Determine the random variable.
- Determine the distribution for the test.
- Draw a graph and calculate the test statistic.
- Compare the calculated test statistic with the Z critical value determined by the level of significance required by the test and make a decision (cannot reject \(H_0\) or cannot accept \(H_0\) ), and write a clear conclusion using English sentences.
Comparing Two Independent Population Means
Two population means from independent samples where the population standard deviations are not known
- Random Variable: \(\bar{X}_1-\bar{X}_2=\) the difference of the sampling means
- Distribution: Student's t-distribution with degrees of freedom (variances not pooled)
Cohen's Standards for Small, Medium, and Large Effect Sizes
Cohen's d is a measure of “effect size” based on the differences between two means.
It is important to note that Cohen's d does not provide a level of confidence as to the magnitude of the size of the effect comparable to the other tests of hypothesis we have studied. The sizes of the effects are simply indicative.
Test for Differences in Means: Assuming Equal Population Variances
In situations when we do not know the population variances but assume the variances are the same, the pooled sample variance will be smaller than the individual sample variances.
This will give more precise estimates and reduce the probability of discarding a good null.
Comparing Two Independent Population Proportions
Test of two population proportions from independent samples.
- Random variable: \(\mathrm{p}_A^{\prime}-\mathrm{p}_B^{\prime}=\) difference between the two estimated proportions
- Distribution: normal distribution
Two Population Means with Known Standard Deviations
A hypothesis test of two population means from independent samples where the population standard deviations are known will have these characteristics:
- Random variable: \(\bar{X}_1-\bar{X}_2=\) the difference of the means
- Distribution: normal distribution
Matched or Paired Samples
A hypothesis test for matched or paired samples (t-test) has these characteristics:
- Test the differences by subtracting one measurement from the other measurement
- Random Variable: \(\bar{x}_d=\) mean of the differences
- Distribution: Student’s-t distribution with n – 1 degrees of freedom
- If the number of differences is small (less than 30), the differences must follow a normal distribution.
- Two samples are drawn from the same set of objects.
- Samples are dependent.