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9.11: Chapter 9 Review

  • Page ID
    6092
  • 9.1 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:

    1. Evaluate the null hypothesis, typically denoted with H0. The null is not rejected unless the hypothesis test shows otherwise. The null statement must always contain some form of equality (=, ≤ or ≥)
    2. 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 > ).
    3. If we reject the null hypothesis, then we can assume there is enough evidence to support the alternative hypothesis.
    4. 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.

    9.2 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.

    9.3 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:

    1. 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.
    2. 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: \(np > 5\) and \(nq > n\) where \(n\) is the sample size, \(p\) is the probability of a success, and \(q\) is the probability of a failure.

    9.4 Full Hypothesis Test Examples

    The hypothesis test itself has an established process. This can be summarized as follows:

    1. Determine \(H_0\) and \(H_a\). Remember, they are contradictory.
    2. Determine the random variable.
    3. Determine the distribution for the test.
    4. Draw a graph and calculate the test statistic.
    5. 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.