8-1 Basics of Hypothesis Testing

• A hypothesis is a claim or statement about a property of a population

• A hypothesis test (or test of significance) is a procedure for testing a claim about a property of a population

• The null hypothesis (denoted by H subscript 0) is a statement that the value of a population parameter is equal to some claimed value

• The alternative hypothesis (denoted by H subscript a) is a statement that the parameter has a value that somehow differs from the null hypothesis. For the methods of this chapter, the symbolic form of the alternative hypothesis must use one of these symbols: <, >, not equal to

• Procedure for hypothesis testing:

  • 1. Identify the claim

  • 2. Give symbolic form

  • 3. Identify null and alternative hypothesis (the alternative is the one NOT containing equality)

  • 4. Select significance value (alpha)

  • 5. Identify the test statistic and determine its sampling distribution

    • 6. Find values (p-value or critical value method)

    • 7. Make a decision (p-value or critical value method)

  • 8. Restate decision in nontechnical terms

• The significance level alpha for a hypothesis test is the probability value used as the cutoff for determining when the sample evidence constitutes significant evidence against the null hypothesis. By its nature, the significance level alpha is the probability of mistakenly rejecting the null hypothesis when it is true

• Significance level alpha = P (rejecting H0 when H0 is true)

• The test statistic is a value used in making a decision about the null hypothesis, and it is found by converting the sample statistic to a score with the assumption that the null hypothesis is true

• The critical regionis the area corresponding to all values of the test statistic that cause us to reject the null hypothesis

  • two tailed test: critical region is in 2 extreme regions

  • left-tailed test: critical region is in extreme left region

  • right-tailed test: critical region is in extreme right region

• The p-value is the probability of getting a value of the test statistic that is at least as extreme as the test statistic obtained from the sample data, assuming that the null hypothesis is true

• In a hypothesis test, the critical value(s) separates the critical region (where we reject the null hypothesis) from the values of the test statistic that do not lead to the rejection of the null hypothesis

• When concluding, always make sense of the conclusion with a statement that uses simple nontechnical wording that addresses the original claim

• Type I error: the mistake of rejecting the null hypothesis when it is actually true, equal to alpha

• Type II error: the mistake of failing to reject the null hypothesis when it is actually false, equal to beta

• The power of a hypothesis test is the probability 1-beta of rejecting a false null hypothesis

8-2 Testing a Claim About a Proportion

• When testing claims about proportions, the CI method is not equivalent to the P-value and critical value methods, so the CI method could result in a different conclusion

• With the exact method, the actual probability of a type I error is less than or equal to alpha, which is the desired probability of a type I error

8-3 Testing a Claim About a Mean

• This t test is robust against a departure from normality, meaning the test works reasonably well if the departure from normality is not too extreme

• If suitable technology is available, the p-value method of testing hypotheses is ideal.

8-4 Testing a Claim About a Standard Deviation or Variance

• The chi-square test of this section is not robust against a departure from normality, meaning that the test does not work well if the population has a distribution that is far from normal