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Hypothesis Testing: Types, Steps, and Z-Tests

This chapter covers the basics of hypothesis testing, including types of hypotheses, steps in hypothesis testing, and Z-tests for a population mean with known standard deviation. Learn how to set up hypotheses, calculate test statistics, interpret p-values, and make conclusions.

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Hypothesis Testing: Types, Steps, and Z-Tests

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  1. Chapter 9 Hypothesis Testing

  2. Hypothesis • Hypothesis is a statement about a certain population parameter. • It is usually about a parameter equal to (not equal to), less than or equal to ( greater than), greater than or equal to(or less than) some given number. • For example: the supporting rate of a candidate among all voters is greater than 50%. • A hypothesis testing is a statistical technique for evaluating if there is enough evidence to support a hypothesis. The strength of evidence is evaluated by the probability of certain statistic, called test statistic.

  3. Rule of Rare Events • If under current assumption, the probability to observe the sample (statistic) we have is extremely small, then we conclude the assumption is not right. • How small is small? if the probability to observe the sample is less than the Level of significance . =0.1, .05, .01 It is determined by the risk requirement of the problem. The risk is the probability to make a mistake.

  4. Rule of Events, final version • If under current assumption (H-null), the probability to observe the test statistic is  , then we conclude the assumption is not right. • If under H-null, the p-value is  , then we conclude H-null assumption is not right.

  5. How to set up the symbolic form of the hypothesis-express relationship with math expressions • Find out the hypothesis of interest about the population parameter (mean), and write down its symbolic expression. • Write down the symbolic expression of the complement. • The expression with equal sign (=, ≤, ≥) is called H-null hypothesis (denoted by H0 ), and the remaining expression not containing equal sign is called the H-A hypothesis (denoted by Ha).

  6. Types of Hypotheses • Right-Sided tailed, “Greater Than” Alternative H0: m0 vs. Ha: >m0 • Left-Sided tailed, “Less Than” Alternative H0 :  m0 vs. Ha : <m0 • Two-Sided tailed, “Not Equal To” Alternative H0 :  = m0 vs. Ha : m0 where m0 is a given constant value (with the appropriate units) that is a comparative value

  7. Types of Hypotheses-the red part is the conversion followed by some textbooks • Right-Sided (Right-Tailed), “Greater Than” Alternative H0: m0 vs. Ha: >m0 H0:  = m0 vs. Ha: >m0 • Left-Sided (Left-Tailed), “Less Than” Alternative H0 :  m0 vs. Ha : <m0 H0 :  = m0 vs. Ha : <m0 • Two-Sided (Two-Tailed) , “Not Equal To” Alternative H0 :  = m0 vs. Ha : m0 wherem0is a given constant value (with the appropriate units) that is a comparative value

  8. Z Tests about a Population Mean:σ known • The population standard deviation σ is known. • Suppose the population being sampled is normally distributed, or sample size n is at least 30. Under these two conditions, use the Z distribution to calculate the p-value and then use the rule of rare events to perform the hypothesis testing.

  9. Z Test Statistic-σ is known • Use the “test statistic” • If the population is normal or n is large*, the test statistic t.s. follows a normal distribution *n≥ 30, by the Central Limit Theorem

  10. Z Tests about a Population Mean:σ known

  11. Right-tailed Test • P is the area to the right of the test statistic. • 0 • 1 • 2 • 3 • -3 • -2 • -1 • z • Test statistic • Right-tailed test: P(Z>t.s.) • H0: μ≤k • Ha: μ> k

  12. Left-tailed Test • P is the area to the left of the test statistic. • z • -3 • -2 • -1 • 0 • 1 • 2 • 3 • Test statistic • Left-tailed test: P(Z<t.s.) • H0: μk • Ha: μ< k

  13. Two-tailed Test • P is twice the area to the left of the negative test statistic. • P is twice the area to the right of the positive test statistic. • 0 • 1 • 2 • 3 • -3 • -2 • -1 • z • Test statistic • Test statistic Two-tailed test: 2*P(Z>t.s.) for t.s.>0 2*P(Z<t.s.) for t.s.<0 • H0: μ= k • Ha: μ k

  14. Hypothesis Testing Conclusion • If p-value≤, then we say we reject H-null and accept Ha. • If p-value>, we say we fail to reject H-null and do not accept Ha. Never say we accept H-null.

  15. Z Tests about a Population Mean:σ known, rejection region method • To use the rule of rare events we only need to know the relationship between p-value and the given significance level. See slide 3. • The rejection region method explores the property and sets up rejection regions in which any value corresponds to a p-value less than the given significance level . That means if the t.s. is on the rejection region then we reject H0.

  16. Za and Right Hand Tail Areas • The definition of the critical value Zα • The area to the right if 1-α • Zα • Zαis the percentile such that • P(Z< Zα) =1-α

  17. Right-tailed Test, rejection region • The area to the left of za isα. • 0 • 1 • 2 • 3 • -3 • -2 • -1 • z • Right-tailed test, for any given significance level  • H0: μ≤k • Ha: μ> k za • Test statistic

  18. Left-tailed Test, rejection region • The area to the left of -za isα. • z • -3 • -2 • -1 • 0 • 1 • 2 • 3 • Left-tailed test • H0: μk • Ha: μ< k • Test statistic -za

  19. Two-tailed Test, rejection region The area to the left of -za/2 is α/2. • The area to the right of za/2 isα/2. • 0 • 1 • 2 • 3 • -3 • -2 • -1 • z • Two-tailed test • H0: μ= k • Ha: μ k -za/2 za/2

  20. Z Tests about a Population Mean:σ known, rejection region method

  21. t Tests about a Population Mean:σ Unknown • The population standard deviation σ is unknown, as is the usual situation, but the sample standard deviation s is given. • The population being sampled is normally distributed or sample size is n≥30. • Under these two conditions, we can use the t distribution to test hypotheses

  22. Defining the t Statistic: σ Unknown • Let x be the mean of a sample of size n with standard deviation s • Also, µ0 is the claimed value of the population mean • Define a new test statistic • If the population being sampled is normal or sample size is big enough, and s is given… • The sampling distribution of the t.s. is a t distribution with n – 1 degrees of freedom

  23. t Tests about a Population Mean:σ Unknown Continued

  24. Definition of the critical value tα

  25. Type I and Type II errors • If we reject H0, then it is possible to make type I error • If we fail to reject H0 and do not accept Ha (or equivalently: fail to reject H0 and reject Ha), then it is possible to make type II error.

  26. Selecting an Appropriate Test Statistic for a Test about a Population Mean

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