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Chapter 13

Chapter 13. Introduction to Inference. What We’ll Learn?. <Part I> Confidence Interval Confidence level Margin of error Critical value Z* <Part II> Null hypothesis H 0 Alternative hypothesis H a Z test statistic P Value Significance level . Statistical Inference.

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Chapter 13

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  1. Chapter 13 Introduction to Inference Chapter 13

  2. What We’ll Learn? <Part I> • Confidence Interval • Confidence level • Margin of error • Critical value Z* <Part II> • Null hypothesis H0 Alternative hypothesis Ha • Z test statistic • P Value • Significance level  Chapter 13

  3. Statistical Inference • Provides methods for drawing conclusions about a population from sample data • Confidence Intervals • What is the population mean? • Gives an estimated range of value which is likely to include an unknown population parameter such as population mean µ Chapter 13

  4. Inference about a MeanSimple Conditions ◙ SRS from the population of interest ◙ Variable has a Normal distribution N(m, s) in the population ◙ Although the value of m is unknown, the value of the population standard deviation s is known Chapter 13

  5. Confidence Interval A level C confidence interval has two parts • An confidence interval usually has the form:estimate ± margin of error • The confidence level C, which is the probability that the interval will capture the true parameter value in repeated samples; that is, C is the success rate for the method. Chapter 13

  6. Video references • http://www.youtube.com/watch?v=Ohz-PZqaMtk • http://www.youtube.com/watch?v=Q6Lj_8yt4Qk <short, brief define Confidence interval> • http://www.youtube.com/watch?v=U59Rbpus824 Chapter 13

  7. Case Study NAEP Quantitative Scores (National Assessment of Educational Progress) Rivera-Batiz, F. L., “Quantitative literacy and the likelihood of employment among young adults,” Journal of Human Resources, 27 (1992), pp. 313-328. What is the average score for all young adult males? Chapter 13

  8. Case Study NAEP Quantitative Scores The NAEP survey includes a short test of quantitative skills, covering mainly basic arithmetic and the ability to apply it to realistic problems. Scores on the test range from 0 to 500, with higher scores indicating greater numerical abilities. It is known that NAEP scores have standard deviation s = 60. Chapter 13

  9. Case Study NAEP Quantitative Scores In a recent year, 840 men 21 to 25 years of age were in the NAEP sample. Their mean quantitative score was 272. On the basis of this sample, estimate the mean score m in the population of all 9.5 million young men of these ages. Chapter 13

  10. Case Study NAEP Quantitative Scores To estimate the unknown population mean m, use the sample mean = 272. The law of large numbers suggests that will be close to m, but there will be some error in the estimate. The sampling distribution of has the Normal distribution with mean m and standard deviation Chapter 13

  11. Case Study NAEP Quantitative Scores Chapter 13

  12. The 68-95-99.7 rule indicates that and m are within two standard deviations (4.2) of each other in about 95% of all samples. Case Study NAEP Quantitative Scores Chapter 13

  13. So, if we estimate that m lies within 4.2 of , we’ll be right about 95% of the time. Case Study NAEP Quantitative Scores Chapter 13

  14. Confidence IntervalMean of a Normal Population Take an SRS of size n from a Normal population with unknown mean m and known standard deviation s. A level C confidence interval for m is: z* is called the critical value, and z* and –z* mark off the Central area C under a standard normal curve (next slide); values of z* for many choices of C can be found at the bottom of Table C in the back of the textbook, and the most common values are on the next slide. Chapter 13

  15. Confidence IntervalMean of a Normal Population Chapter 13

  16. Case Study NAEP Quantitative Scores Using the 68-95-99.7 rule gave an approximate 95% confidence interval. A more precise 95% confidence interval can be found using the appropriate value of z* (1.960) with the previous formula. We are 95% confident that the average NAEP quantitative score for all adult males is between 267.884 and 276.116. Chapter 13

  17. Express Confidence Interval • The confidence level describes the uncertainty associated with a sampling method • A 95% confidence level means that we would expect 95% of the interval estimates to include the population parameter • To express a confidence interval, you need three pieces of information. ◙ Confidence level ◙ Statistic ◙ Margin of error Chapter 13

  18. Construct A Confidence Interval • Identify a sample statistic. Locate the statistic (e.g, mean, standard deviation) • Select a confidence level, choose 90%, 95%, or 99% confidence levels or others • Find the margin of error: Margin of error = Critical value x Standard deviation of statistic • Confidence interval = sample statistic + margin of error Chapter 13

  19. Exercise 1 • Suppose we want to estimate the average weight of an adult male in Dekalb County, GA. We draw a random sample of 1,000 men from a population of 1,000,000 men and weigh them. We find that the average weighs in sample is 180 pounds, and the standard deviation is 30 pounds. What is the 95% confidence interval. • (A) 180 + 1.86 (B) 180 + 3.0 (C) 180 + 5.88 (D) 180 + 30 Chapter 13

  20. Hypothesis Test & P Value • Hypothesis, H0, Ha • Significance • Sample • P-value • Conclusion From sample mean to draw an conclusion about population mean Chapter 13

  21. Stating HypothesesNull Hypothesis, H0 • The statement being tested in a statistical test is called 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 13

  22. 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 is what we are trying to prove. Chapter 13

  23. 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 13

  24. 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 13

  25. 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 13

  26. Case Study I Sweetening Colas Chapter 13

  27. 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 13

  28. 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 13

  29. 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 13

  30. 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 13

  31. 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 13

  32. 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 13

  33. P-value ■ The p-value is the probability or area marked by the teststatistic z (from sample) in the normal distribution curve when assuming that the null hypothesis is true. ■ The smaller the p-value, the more significant is the difference between the null hypothesis and the sample results. ■ The smaller the P-value, the stronger the evidence the data provide against the null hypothesis. Chapter 13

  34. P-value for Testing Means • Ha: m > m0 when Ha contains “greater than” symbol, Perform aRight- tailed test • P-value is the probability of getting a value as large or larger than the observed test statistic (z) value. • Ha: m < m0when Ha contains “less than” symbol, Perform a Left-tailed test • P-value is the probability of getting a value as small or smaller than the observed test statistic (z) value. • Ha: m ¹ m0 when Ha contains “not equal to” symbol, Perform aTwo-tailed test • 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 13

  35. Chapter 13

  36. 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 13

  37. Case Study I Sweetening Colas Chapter 13

  38. 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 13

  39. 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 13

  40. Case Study II Studying Job Satisfaction Chapter 13

  41. Statistical Significance α • 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. • “Rejects the null hypothesis" when the p-value is less than the significance level α • 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 13

  42. Tests Procedure for a Population Mean The five steps in carrying out a significance test: • State the null and alternative hypotheses. • Set the level of significance, usually ɑ = 0.05 • Take a sample from population and provide the Z test statistic. • Locate the P-value from Table A. • Using P-value compare with ɑ to reject null hypothesis if P-value < ɑ Chapter 13

  43. 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 13

  44. 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 13

  45. 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 13

  46. Interesting Video http://www.youtube.com/watch?v=l9ueYYpYU_s <Tailed test> http://www.youtube.com/watch?v=BX9iMIC6mcg <Five Steps> http://www.youtube.com/watch?v=cW16A7hXbTo <P-Value> Chapter 13

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