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Hypothesis Testing: Understanding Steps and Definitions

Learn the steps and definitions used in hypothesis testing, including the null and alternative hypotheses, finding critical values, and determining the level of significance. Practice problems and solutions provided.

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Hypothesis Testing: Understanding Steps and Definitions

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  1. Chapter 8 Hypothesis Testing

  2. Section 8-1: Steps in Hypothesis Testing – Traditional Method • Learning targets • IWBAT understand the definitions used in hypothesis testing. • IWBAT state the null and alternative hypotheses. • IWBAT find critical values for the z-test

  3. Vocabulary • Statistical hypothesis – a conjecture about a population parameter. This conjecture may or may not be true. • Null hypothesis – symbolized as H0, is a statistical hypothesis that states that there is no difference between a parameter and a specific value, or that there is no difference between two parameters. • Alternative hypothesis – symbolized as H1, is a statistical hypothesis that states the existence of a difference between a parameter and a specific value, or states that there is a difference between two parameters.

  4. Practice Problems

  5. Solutions

  6. A statistical test uses the data obtained from a sample to make a decision about whether the null hypothesis should be rejected. • The numerical value obtained from a statistical test is called the test value.

  7. Errors In hypothesis testing there are 2 types of errors: • Type I Error – you reject the null hypothesis when it is true • Type II Error – you do not reject the null hypothesis when it is false Example: jury trial outcomes

  8. Level of Significance • represented by alpha (α) • the value used to determine the critical value that helps determine whether or not to reject the null hypothesis • Also referred to as the P-value which is the area under the curve

  9. Critical Value and Region Critical value – the z-value that separates the critical region from the noncritical region (symbol C.V.) Critical/rejection region – range of values of the test value that indicates that there is a significant difference and that the null hypothesis should be rejected Noncritical/nonrejection region – range of values of the test value that indicates that the difference was probably due to chance and the null hypothesis should not be rejected

  10. One-Tailed Test

  11. Two-Tailed Test

  12. This chart contains the z-scores for the most used α The z-scores are found the same way they were in Section 6-1.

  13. Practice Problems

  14. Section 8-2 Z Test for a Mean

  15. Steps • State null and alternative hypotheses. • Find the critical values • Compute the test value • Make decision to reject or accept • Summarize the results

  16. Critical Value and Region Critical value – the z-value that separates the critical region from the noncritical region (symbol C.V.) Critical/rejection region – range of values of the test value that indicates that there is a significant difference and that the null hypothesis should be rejected Noncritical/nonrejection region – range of values of the test value that indicates that the difference was probably due to chance and the null hypothesis should not be rejected

  17. This chart contains the z-scores for the most used α The z-scores are found the same way they were in Section 6-1.

  18. Formula X = value = mean = Standard deviation n = sample size

  19. One-Tailed Test

  20. Two-Tailed Test

  21. Summarize Results • To summarize the results you need to state whether there is or is not sufficient evidence to support the claim (the alternative hypothesis) - If we reject the null there is sufficient evidence to support the claim - If we fail to reject the null there is not sufficient evidence to support the claim. Example: There is sufficient evidence to support the claim that students will have an average score of 19 on the ACT.

  22. Therefore, our decision is to reject the null hypothesis. Test Statistic Thus, there is sufficient evidence to support the claim that the valve does not perform to specifications. Two-tailed with α=.05, therefore critical region starts at 1.96. Since the situation is two tailed, we have a tail to the right and a tail to the left. If we compare the two z-scores, we notice that the test statistic is greater than the critical value.

  23. Since we have rejected the null, we can conclude there is sufficient evidence to support the claim that the state employees earn on average less than the federal employees. Test Statistic Since the claim is “less than” the situation is one-tailed. The z-score critical value for α=.01 is -2.33 When we compare the two z-scores, we notice that the test statistic is less than the critical value and falls in the rejection region. Therefore, we will reject the null hypothesis.

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