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Understanding Hypothesis Testing in Statistics

Explore the basics of hypothesis testing, including stating null and alternative hypotheses, conducting significance tests, and interpreting results in various case studies.

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Understanding Hypothesis Testing in Statistics

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  1. Chapter 14 Tests of Significance: The Basics Chapter 14

  2. Reasoning of Tests of Significance • What would happen if we repeated the sample or experiment many times? • How likely would it be to see the results we saw if the claim of the test were true? • Do the data give evidence against the claim? Chapter 14

  3. Stating HypothesesNull Hypothesis, H0 • The statement being tested in a statistical test is called the null hypothesis. • The test is designed to assess the strength of evidence against the null hypothesis. • Usually the null hypothesis is a statement of “no effect” or “no difference”, or it is a statement of equality. • When performing a hypothesis test, we assume that the null hypothesis is true until we have sufficient evidence against it. Chapter 14

  4. Stating HypothesesAlternative Hypothesis, Ha • The statement we are trying to find evidence for is called the alternative hypothesis. • Usually the alternative hypothesis is a statement of “there is an effect” or “there is a difference”, or it is a statement of inequality. • The alternative hypothesis should express the hopes or suspicions we bring to the data. It is cheating to first look at the data and then frame Ha to fit what the data show. Chapter 14

  5. Case Study I Sweetening Colas Diet colas use artificial sweeteners to avoid sugar. These sweeteners gradually lose their sweetness over time. Trained testers sip the cola and assign a “sweetness score” of 1 to 10. The cola is then retested after some time and the two scores are compared to determine the difference in sweetness after storage. Bigger differences indicate bigger loss of sweetness. Chapter 14

  6. Case Study I Sweetening Colas Suppose we know that for any cola, the sweetness loss scores vary from taster to taster according to a Normal distribution with standard deviation s = 1. The mean m for all tasters measures loss of sweetness. The sweetness losses for a new cola, as measured by 10 trained testers, yields an average sweetness loss of = 1.02. Do the data provide sufficient evidence that the new cola lost sweetness in storage? Chapter 14

  7. Case Study I Sweetening Colas • If the claim that m = 0 is true (no loss of sweetness, on average), the sampling distribution of from 10 tasters is Normal with m = 0 and standard deviation • The data yielded = 1.02, which is more than three standard deviations from m = 0. This is strong evidence that the new cola lost sweetness in storage. • If the data yielded = 0.3, which is less than one standard deviations from m = 0, there would be no evidence that the new cola lost sweetness in storage. Chapter 14

  8. Case Study I Sweetening Colas Chapter 14

  9. The Hypotheses for Means • Null: H0:m= m0 • One sided alternatives Ha:m > m0 Ha:m < m0 • Two sided alternative Ha:m ¹ m0 Chapter 14

  10. Case Study I Sweetening Colas The null hypothesis is no average sweetness loss occurs, while the alternative hypothesis (that which we want to show is likely to be true) is that an average sweetness loss does occur. H0: m = 0 Ha: m > 0 This is considered a one-sided test because we are interested only in determining if the cola lost sweetness (gaining sweetness is of no consequence in this study). Chapter 14

  11. Case Study II Studying Job Satisfaction Does the job satisfaction of assembly workers differ when their work is machine-paced rather than self-paced? A matched pairs study was performed on a sample of workers, and each worker’s satisfaction was assessed after working in each setting. The response variable is the difference in satisfaction scores, self-paced minus machine-paced. Chapter 14

  12. Case Study II Studying Job Satisfaction The null hypothesis is no average difference in scores in the population of assembly workers, while the alternative hypothesis (that which we want to show is likely to be true) is there is an average difference in scores in the population of assembly workers. H0: m = 0 Ha: m ≠ 0 This is considered a two-sided test because we are interested determining if a difference exists (the direction of the difference is not of interest in this study). Chapter 14

  13. Test StatisticTesting the Mean of a Normal Population Take an SRS of size n from a Normal population with unknown mean m and known standard deviation s. The test statistic for hypotheses about the mean (H0: m = m0) of a Normal distribution is the standardized version of : Chapter 14

  14. Case Study I Sweetening Colas If the null hypothesis of no average sweetness loss is true, the test statistic would be: Because the sample result is more than 3 standard deviations above the hypothesized mean 0, it gives strong evidence that the mean sweetness loss is not 0, but positive. Chapter 14

  15. P-value Assuming that the null hypothesis is true, the probability that the test statistic would take a value as extreme or more extreme than the value actually observed is called the P-value of the test. The smaller the P-value, the stronger the evidence the data provide against the null hypothesis. That is, a small P-value indicates a small likelihood of observing the sampled results if the null hypothesis were true. Chapter 14

  16. P-value for Testing Means • Ha: m > m0 • P-value is the probability of getting a value as large or larger than the observed test statistic (z) value. • Ha: m < m0 • P-value is the probability of getting a value as small or smaller than the observed test statistic (z) value. • Ha: m ¹ m0 • P-value is two times theprobability of getting a value as large or larger than the absolute value of the observed test statistic (z) value. Chapter 14

  17. Case Study I Sweetening Colas For test statistic z = 3.23 and alternative hypothesisHa: m > 0, the P-value would be: P-value = P(Z > 3.23) = 1 – 0.9994 = 0.0006 If H0 is true, there is only a 0.0006 (0.06%) chance that we would see results at least as extreme as those in the sample; thus, since we saw results that are unlikely if H0 is true, we therefore have evidence against H0 and in favor of Ha. Chapter 14

  18. Case Study I Sweetening Colas Chapter 14

  19. Case Study II Studying Job Satisfaction Suppose job satisfaction scores follow a Normal distribution with standard deviation s = 60. Data from 18 workers gave a sample mean score of 17. If the null hypothesis of no average difference in job satisfaction is true, the test statistic would be: Chapter 14

  20. Case Study II Studying Job Satisfaction For test statistic z = 1.20 and alternative hypothesisHa: m ≠ 0, the P-value would be: P-value = P(Z < -1.20 or Z > 1.20) = 2 P(Z < -1.20) = 2 P(Z > 1.20) = (2)(0.1151) = 0.2302 If H0 is true, there is a 0.2302 (23.02%) chance that we would see results at least as extreme as those in the sample; thus, since we saw results that are likely if H0 is true, we therefore do not have good evidence against H0 and in favor of Ha. Chapter 14

  21. Case Study II Studying Job Satisfaction Chapter 14

  22. Statistical Significance • If the P-value is as small as or smaller than the significance level a (i.e., P-value ≤ a), then we say that the data give results that are statistically significant at level a. • If we choose a = 0.05, we are requiring that the data give evidence against H0 so strong that it would occur no more than 5% of the time when H0 is true. • If we choose a = 0.01, we are insisting on stronger evidence against H0, evidence so strong that it would occur only 1% of the time when H0 is true. Chapter 14

  23. Tests for a Population Mean The four steps in carrying out a significance test: • State the null and alternative hypotheses. • Calculate the test statistic. • Find the P-value. • State your conclusion in the context of the specific setting of the test. The procedure for Steps 2 and 3 is on the next page. Chapter 14

  24. Chapter 14

  25. Case Study I Sweetening Colas • Hypotheses: H0: m = 0 Ha: m > 0 • Test Statistic: • P-value: P-value = P(Z > 3.23) = 1 – 0.9994 = 0.0006 • Conclusion: Since the P-value is smaller than a = 0.01, there is very strong evidence that the new cola loses sweetness on average during storage at room temperature. Chapter 14

  26. Case Study II Studying Job Satisfaction • Hypotheses: H0: m = 0 Ha: m ≠ 0 • Test Statistic: • P-value: P-value = 2P(Z > 1.20) = (2)(1 – 0.8849) = 0.2302 • Conclusion: Since the P-value is larger than a = 0.10, there is not sufficient evidence that mean job satisfaction of assembly workers differs when their work is machine-paced rather than self-paced. Chapter 14

  27. Confidence Intervals & Two-Sided Tests A level a two-sided significance test rejects the null hypothesis H0: m = m0 exactly when the value m0 falls outside a level 1 – a confidence interval for m. Chapter 14

  28. Case Study II Studying Job Satisfaction A 90% confidence interval for m is: Since m0 = 0 is in this confidence interval, it is plausible that the true value of m is 0; thus, there is not sufficient evidence(at  = 0.10) that the mean job satisfaction of assembly workers differs when their work is machine-paced rather than self-paced. Chapter 14

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