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Chapter 22

Chapter 22. What Is a Test of Significance?. Thought Question 1.

violetta-michel
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Chapter 22

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  1. Chapter 22 What Is a Test of Significance? Chapter 22

  2. Thought Question 1 The defendant in a court case is either guilty or innocent. Which of these is assumed to be true when the case begins? The jury looks at the evidence presented and makes a decision about which of these two options appears more plausible. Depending on this decision, what are the two types of errors that could be made by the jury? Which is more serious? Chapter 22

  3. Thought Question 2 Suppose 60% (0.60) of the population are in favor of new tax legislation. A random sample of 265 people results in 175, or 0.66, who are in favor. From the Rule for Sample Proportions, we know the potential sample proportions in this situation follow an approximately normal distribution, with a mean of 0.60 and a standard deviation of 0.03. Find the standardized score for the observed value of 0.66; then find the probability of observing a standardized score at least that large or larger. Chapter 22

  4. 0.51 0.54 0.57 0.60 0.63 0.66 0.69 Thought Question 2: Bell-Shaped Curve of Sample Proportions (n=265) mean = 0.60 S.D. = 0.03 2.27% Chapter 22

  5. Thought Question 3 Suppose that in the previous question we do not know for sure that the proportion of the population who favor the new tax legislation is 60%. Instead, this is just the claim of a politician. From the data collected, we have discovered that if the claim is true, then the sample proportion observed falls at the 97.73 percentile (about the 98th percentile) of possible sample proportions for that sample size. Should we believe the claim and conclude that we just observed strange data, or should we reject the claim? What if the result fell at the 85th percentile? At the 99.99th percentile? Chapter 22

  6. 99.99th 98th 85th 0.51 0.54 0.57 0.60 0.63 0.66 0.69 Thought Question 3: Bell-Shaped Curve of Sample Proportions (n=265) Chapter 22

  7. Case Study Parental Discipline Brown, C. S., (1994) “To spank or not to spank.” USA Weekend, April 22-24, pp. 4-7. What are parents’ attitudes and practices on discipline? Chapter 22

  8. Case Study: Survey Parental Discipline • Nationwide random telephone survey of 1,250 adults. • 474 respondents had children under 18 living at home • results on behavior based on the smaller sample • reported margin of error • 3% for the full sample • 5% for the smaller sample Chapter 22

  9. Case Study: Results Parental Discipline “The 1994 survey marks the first time a majority of parents reported not having physically disciplined their children in the previous year. Figures over the past six years show a steady decline in physical punishment, from a peak of 64 percent in 1988” • The 1994 sample proportion who did not spank or hit was 51% ! • Is this evidence that a majority of the population did not spank or hit? Chapter 22

  10. The Five Steps of Hypothesis Testing • Determining the Two Hypotheses • Computing the Sampling Distribution • Collecting and Summarizing the Data(calculating the observed test statistic) • Determining How Unlikely the Test Statistic is if the Null Hypothesis is True (calculating the P-value) • Making a Decision/Conclusion(based on the P-value, is the result statistically significant?) Chapter 22

  11. The Null Hypothesis: H0 • population parameter equals some value • status quo • no relationship • no change • no difference in two groups • etc. • When performing a hypothesis test, we assume that the null hypothesis is true until we have sufficient evidence against it Chapter 22

  12. The Alternative Hypothesis: Ha • population parameter differs from some value • not status quo • relationship exists • a change occurred • two groups are different • etc. Chapter 22

  13. The Hypotheses for Proportions • Null: H0:p=p0 • One sided alternatives Ha:p>p0 Ha:p<p0 • Two sided alternative Ha:p¹p0 Chapter 22

  14. Case Study: The Hypotheses • Null: The proportion of parents who physically disciplined their children in the previous year is the same as the proportion [p] of parents who did not physically discipline their children. [H0: p=.5] • Alt: A majority of parents did not physically discipline their children in the previous year. [Ha: p>.5] Chapter 22

  15. If numerous simple random samples of size n are taken, the sample proportions from the various samples will have an approximately normaldistribution with mean equal to p (the population proportion) and standard deviation equal to Sampling Distribution for Proportions Since we assume the null hypothesis is true, we replace p with p0 to complete the test. Chapter 22

  16. Test Statistic for Proportions To determine if the observed proportion is unlikely to have occurred under the assumption that H0 is true, we must first convert the observed value to a standardized score: Chapter 22

  17. Case Study: Test Statistic • Based on the sample: • n=474 (large, so proportions follow normal distribution) • no physical discipline: 51% • standard error of : • (where .50 is p0 from the null hypothesis) • standardized score (test statistic) • z = (0.51 - 0.50) / 0.023 = 0.43 Chapter 22

  18. P-value • The P-value is the probability of observing data this extreme or more so in a sample of this size, assuming that the null hypothesis is true. • A small P-value indicates that the observed data (or relationship) is unlikely to have occurred if the null hypothesis were actually true • The P-value tends to be small when there is evidence in the data against the null hypothesis • The P-value is NOT the probability that the null hypothesis is true Chapter 22

  19. P-value for Testing Proportions • Ha: p>p0 • When the alternative hypothesis includes a greater than “>” symbol, the P-value is the probability of getting a value as large or larger than the observed test statistic (z) value. • look up the percentile for the value of z in the standard normal table (Table B) • the P-value is 1 minus this probability Chapter 22

  20. P-value for Testing Proportions • Ha: p<p0 • When the alternative hypothesis includes a less than “<” symbol, the P-value is the probability of getting a value as small or smaller than the observed test statistic (z) value. • look up the percentile for the value of z in the standard normal table (Table B) • the P-value is this probability Chapter 22

  21. P-value for Testing Proportions • Ha: pp0 • When the alternative hypothesis includes a not equal to “” symbol, the P-value is found as follows: • make the value of the observed test statistic (z) positive (absolute value) • look up the percentile for this positive value of z in the standard normal table (Table B) • find 1 minus this probability • double the answer to get the P-value Chapter 22

  22. Alternative Method for P-value Caution: Use this method only when Ha has a “>” sign and Z is positive, or when Ha has a “<“ sign and Z is negative, or when Ha has a “” sign. • Make the value of the observed test statistic (z) negative • Look up the percentile for this negative value of z in the standard normal table (Table B) • if the alternative hypothesis includes a greater than “>” or less than “<“ symbol, the P-value is this probability in step 2 • if the alternative hypothesis includes a not equal to “” symbol, double this probability in step 2 to get the P-value Chapter 22

  23. 0.51 0.431 0.454 0.477 0.500 0.523 0.546 0.569 -3 -2 -1 0 1 2 3 z: z=0.43 Case Study: P-value Ha: p>.50 P-value = 0.3446 From Table B, z=0.4 is the 65.54th percentile. Chapter 22

  24. * Decision * • If we think the P-value istoo low to believe the observed test statistic is obtained by chance only, then we would reject chance (reject the null hypothesis) and conclude that a statistically significant relationship exists (accept the alternative hypothesis). • Otherwise, we fail to reject chance anddo not reject the null hypothesis of no relationship (result not statistically significant). Chapter 22

  25. Typical Cut-off for the P-value • Commonly, P-values less than 0.05 are considered to be small enough to reject chance (reject the null hypothesis). • Some researchers use 0.10 or 0.01 as the cut-off instead of 0.05. • This “cut-off” value is typically referred to as the significance level  of the test Chapter 22

  26. Case Study: Decision • Since the P-value (.3446) is not small, we cannot reject chance as the reason for the difference between the observed proportion (0.51) and the (null) hypothesized proportion (0.50). • We do not find the result to be statistically significant. • We fail to reject the null hypothesis. It is plausible that there was not a majority (over 50%) of parents who refrained from using physical discipline. Chapter 22

  27. Decision Errors: Type I • If we decide there is a relationship in the population (reject null hypothesis) • This is an incorrect decision only if the null hypothesis is true. • The probability of this incorrect decision is equal to the cut-off () for the P-value. • If the null hypothesis is true and the cut-off is 0.05 • There really is no relationship and the extremity of the test statistic is due to chance. • About 5% of all samples from this population will lead us to wrongly reject chance. Chapter 22

  28. Decision Errors: Type II • If we decide not to reject chance and thus allow for the plausibility of the null hypothesis • This is an incorrect decision only if the alternative hypothesis is true. • The probability of this incorrect decision depends on • the magnitude of the true relationship, • the sample size, • the cut-off for the P-value. Chapter 22

  29. Power of a Test • This is the probability that the sample we collect will lead us to reject the null hypothesis when the alternative hypothesis is true. • The power is larger for larger departures of the alternative hypothesis from the null hypothesis (magnitude of difference) • The power may be increased by increasing the sample size. Chapter 22

  30. Case Study: Decision Error? • Decision: fail to reject H0 • If in the population there truly was a majority of parents who did not physically discipline their children, then we have committed a Type II error. • Could we have committed a Type I error with the decision that we made? [No! Why?] Chapter 22

  31. Key Concepts (1st half of Ch. 22) • Decisions are often made on the basis of incomplete information. • Five Steps of Hypothesis Testing • P-values and Statistical Significance • Decision Errors • Power of a Test Chapter 22

  32. Inference for Population MeansHypothesis Testing The remainder of this chapter discusses the situation when interest is completing hypothesis tests about population means rather than population proportions Chapter 22

  33. Thought Question 4 A study showed that the difference in sample means for the heights of tomato plants when using a nutrient rich potting soil versus using ordinary top soil was 6.82 inches. The corresponding standard error was 3.10 inches. Suppose the means are actually equal, so that the mean difference in heights for the populations is actually zero. What is the standardized score (z) corresponding to the observed difference of 6.82 inches? How often would you expect to see a standardized score that large or larger? Chapter 22

  34. Thought Question 4: Answer standardized score: [(sample mean diff.)  (population mean diff.)] divided by [standard error of the mean difference] z = (6.82  0) / 3.10 = 2.2 This is the 98.61 percentile for a standard normal curve; so the probability of seeing a z-value this large or larger is 1.39% (.0139) Chapter 22

  35. Thought Question 5 One of the conclusions made by researchers from a study comparing the amount of bacteria in carpeted and uncarpeted rooms was, “The average difference [in mean bacteria colonies per cubic foot] was 3.48 colonies (95% CI: -2.72, 9.68; P-value = 0.29).” What are the null and alternative hypotheses being tested here? Is there a statistically significant difference between the means of the two groups? Chapter 22

  36. Thought Question 5: Answer • Null: The mean number of bacteria for carpeted rooms is equal to the mean number of bacteria for uncarpeted rooms. • Alt: The mean number of bacteria for carpeted rooms is different from the mean number of bacteria for uncarpeted rooms. • P-value is large (>.05), so there is not a significant difference (fail to reject the Null hypothesis) [Also, the confidence interval for the difference contains 0.] Chapter 22

  37. Case Study (cont. from Ch. 21) Exercise and Pulse Rates Hypothetical Is the mean resting pulse rate of adult subjects who regularly exercise different from the mean resting pulse rate of those who do not regularly exercise? Use Hypothesis Testing for means Chapter 22

  38. Case Study: Results Exercise and Pulse Rates • Observed difference in means = 75  66 = 9 • Standard error of the difference = 2.26 (given) Chapter 22

  39. The Five Steps of Hypothesis Testing • Determining the Two Hypotheses • Computing the Sampling Distribution • Collecting and Summarizing the Data(calculating the observed test statistic) • Determining How Unlikely the Test Statistic is if the Null Hypothesis is True (calculating the P-value) • Making a Decision/Conclusion(based on the P-value, is the result statistically significant?) Chapter 22

  40. The Hypotheses for aSingle Mean • Null: H0:m=m0 • One sided alternatives Ha:m>m0 Ha:m<m0 • Two sided alternative Ha:m¹m0 Chapter 22

  41. The Hypotheses for aDifference in Two Means • Null: H0:mdiff =mdiff,0(usually = 0) • One sided alternatives Ha:mdiff >mdiff,0 Ha:mdiff <mdiff,0 • Two sided alternative Ha:mdiff ¹mdiff,0 Chapter 22

  42. Case Study: The Hypotheses • Null: The mean resting pulse rate of adult subjects who regularly exercise is the same as the mean resting pulse rate of those who do not regularly exercise? [H0: mdiff =0] • Alt: The mean resting pulse rate of adult subjects who regularly exercise is different from the mean resting pulse rate of those who do not regularly exercise? [Ha: mdiff ¹0] Chapter 22

  43. If numerous simple random samples of size n are taken, the sample means from the various samples will have an approximately normaldistribution with mean equal to m (the population mean) and standard deviation equal to Sampling Distribution for Means Since we assume the null hypothesis is true, we replace m with m0 to complete the test. Also, we again replace the unknown value of  with the sample value s Chapter 22

  44. where is usually 0. Test Statistics for Means Chapter 22

  45. Case Study: Test Statistic • Based on the samples: • large, random samples, so sample means follow normal distribution • observed difference in means = 9 • standard error of the difference = 2.26 • standardized score (test statistic) • z = (9  0) / 2.26 = 3.98 Chapter 22

  46. P-value for Testing Means • Ha: m>m0 • When the alternative hypothesis includes a greater than “>” symbol, the P-value is the probability of getting a value as large or larger than the observed test statistic (z) value. • look up the percentile for the value of z in the standard normal table (Table B) • the P-value is 1 minus this probability Chapter 22

  47. P-value for Testing Means • Ha: m<m0 • When the alternative hypothesis includes a less than “<” symbol, the P-value is the probability of getting a value as small or smaller than the observed test statistic (z) value. • look up the percentile for the value of z in the standard normal table (Table B) • the P-value is this probability Chapter 22

  48. P-value for Testing Means • Ha: mm0 • When the alternative hypothesis includes a not equal to “” symbol, the P-value is found as follows: • make the value of the observed test statistic (z) positive (absolute value) • look up the percentile for this positive value of z in the standard normal table (Table B) • find 1 minus this probability • double the answer to get the P-value Chapter 22

  49. P-value < 20.0003 = 0.0006 9.0 -9.04 -6.78 -4.52 -2.26 0.00 2.26 4.52 6.78 9.04 diff: -4 -3 -2 -1 0 1 2 3 4 z: z=3.98 Case Study: P-value Ha: diff 0 < 0.0003 < 0.0003 From Table B, 3.98 is beyond the 99.97th percentile. Chapter 22

  50. Case Study: Decision • Since the P-value (<.0006) is small, we reject chance as the reason for the difference between the observed means (9). • We find the result to be statistically significant. • We reject the null hypothesis. The data provide evidence that the two population means (resting pulse rates) are not the same. Chapter 22

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