1 Chapter 9: Statistical Inference: Significance Tests About Hypotheses Section 9.1: What Are the Steps for Performing a Significance Test?

2 Learning Objectives 1. 5 Steps of a Significance Test 2. Assumptions 3. Hypotheses 4. Calculate the test statistic 5. P-Value 6. Conclusion and Statistic Significance

3 Learning Objective 1: Significance Test A significance test is a method of using data to summarize the evidence about a hypothesis A significance test about a hypothesis has five steps 1. Assumptions 2. Hypotheses 3. Test Statistic 4. P-value 5. Conclusion

4 Learning Objective 2: Step 1: Assumptions A (significance) test assumes that the data production used randomization Other assumptions may include: Assumptions about the sample size Assumptions about the shape of the population distribution

5 Learning Objective 3: Step 2: Hypothesis A hypothesis is a statement about a population, usually of the form that a certain parameter takes a particular numerical value or falls in a certain range of values The main goal in many research studies is to check whether the data support certain hypotheses

6 Learning Objective 3: Step 2: Hypotheses Each significance test has two hypotheses: The null hypothesis is a statement that the parameter takes a particular value. It has a single parameter value. The alternative hypothesis states that the parameter falls in some alternative range of values.

7 Learning Objective 3: Null and Alternative Hypotheses The value in the null hypothesis usually represents no effect The symbol H o denotes null hypothesis The value in the alternative hypothesis usually represents an effect of some type The symbol H a denotes alternative hypothesis The alternative hypothesis should express what the researcher hopes to show. The hypotheses should be formulated before viewing or analyzing the data!

8 Learning Objective 4: Step 3: Test Statistic A test statistic describes how far the point estimate falls from the parameter value given in the null hypothesis (usually in terms of the number of standard errors between the two). If the test statistic falls far from the value suggested by the null hypothesis in the direction specified by the alternative hypothesis, it is good evidence against the null hypothesis and in favor of the alternative hypothesis. We use the test statistic to assesses the evidence against the null hypothesis by giving a probability, the P-Value.

9 Learning Objective 5: Step 4: P-value To interpret a test statistic value, we use a probability summary of the evidence against the null hypothesis, H o First, we presume that H o is true Next, we consider the sampling distribution from which the test statistic comes We summarize how far out in the tail of this sampling distribution the test statistic falls

10 Learning Objective 5: Step 4: P-value We summarize how far out in the tail the test statistic falls by the tail probability of that value and values even more extreme This probability is called a P-value The smaller the P-value, the stronger the evidence is against H o

11 Learning Objective 5: Step 4: P-value

12 Learning Objective 5: Step 4: P-value The P-value is the probability that the test statistic equals the observed value or a value even more extreme It is calculated by presuming that the null hypothesis H is true 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.

13 Learning Objective 6: Step 5: Conclusion The conclusion of a significance test reports the P-value and interprets what it says about the question that motivated the test

14 Chapter 9: Statistical Inference: Significance Tests About Hypotheses Section 9.2: Significance Tests About Proportions

15 Learning Objectives: 1. Steps of a Significance Test about a Population Proportion 2. Example: One-Sided Hypothesis Test 3. How Do We Interpret the P-value? 4. Two-Sided Hypothesis Test for a Population Proportion 5. Summary of P-values for Different Alternative Hypotheses 6. Significance Level 7. One-Sided vs Two-Sided Tests 8. The Binomial Test for Small Samples

16 Learning Objective 1: Example: Are Astrologers’ Predictions Better Than Guessing? An astrologer prepares horoscopes for 116 adult volunteers. Each subject also filled out a California Personality Index (CPI) survey. For a given adult, his or her horoscope is shown to the astrologer along with their CPI survey as well as the CPI surveys for two other randomly selected adults. The astrologer is asked which survey is the correct one for that adult With random guessing, p = 1/3 The astrologers’ claim: p > 1/3 The hypotheses for this test:  H o : p = 1/3  H a : p > 1/3

17 Learning Objective 1: Steps of a Significance Test about a Population Proportion Step 1: Assumptions The variable is categorical The data are obtained using randomization The sample size is sufficiently large that the sampling distribution of the sample proportion is approximately normal: np ≥ 15 and n(1-p) ≥ 15

18 Learning Objective 1: Steps of a Significance Test about a Population Proportion Step 2: Hypotheses The null hypothesis has the form: H 0 : p = p 0 The alternative hypothesis has the form: H a : p > p 0 (one-sided test) or H a : p < p 0 (one-sided test) or H a : p ≠ p 0 (two-sided test)

19 Learning Objective 1: Steps of a Significance Test about a Population Proportion Step 3: Test Statistic The test statistic measures how far the sample proportion falls from the null hypothesis value, p 0, relative to what we’d expect if H 0 were true The test statistic is:

20 Learning Objective 1: Steps of a Significance Test about a Population Proportion Step 4: P-value The P-value summarizes the evidence It describes how unusual the observed data would be if H 0 were true

21 Learning Objective 1: Steps of a Significance Test about a Population Proportion Step 5: Conclusion We summarize the test by reporting and interpreting the P-value

22 Learning Objective 2: Example 1 Step 1: Assumptions The data is categorical – each prediction falls in the category “correct” or “incorrect” Each subject was identified by a random number. Subjects were randomly selected for each experiment. np=116(1/3) > 15 n(1-p) = 116(2/3) > 15

23 Learning Objective 2: Example 1 Step 2: Hypotheses H 0: p = 1/3 H a: p > 1/3

24 Learning Objective 2: Example 1 Step 3: Test Statistic: In the actual experiment, the astrologers were correct with 40 of their 116 predictions (a success rate of 0.345)

25 Learning Objective 2: Example 1 Step 4: P-value The P-value is 0.40

26 Learning Objective 2: Example 1 Step 5: Conclusion The P-value of 0.40 is not especially small It does not provide strong evidence against H 0: p = 1/3 There is not strong evidence that astrologers have special predictive powers

27 Learning Objective 3: How Do We Interpret the P-value? A significance test analyzes the strength of the evidence against the null hypothesis We start by presuming that H 0 is true The burden of proof is on H a

28 Learning Objective 3: How Do We Interpret the P-value? The approach used in hypothesis testing is called a proof by contradiction To convince ourselves that H a is true, we must show that data contradict H 0 If the P-value is small, the data contradict H 0 and support H a

29 Learning Objective 4: Two-Sided Significance Tests A two-sided alternative hypothesis has the form H a : p ≠ p 0 The P-value is the two-tail probability under the standard normal curve We calculate this by finding the tail probability in a single tail and then doubling it

30 Learning Objective 4: Example 2 Study: investigate whether dogs can be trained to distinguish a patient with bladder cancer by smelling compounds released in the patient’s urine

31 Learning Objective 4: Example 2 Experiment: Each of 6 dogs was tested with 9 trials In each trial, one urine sample from a bladder cancer patient was randomly place among 6 control urine samples

32 Learning Objective 4: Example 2 Results: In a total of 54 trials with the six dogs, the dogs made the correct selection 22 times (a success rate of 0.407)

33 Learning Objective 4: Example 2 Does this study provide strong evidence that the dogs’ predictions were better or worse than with random guessing?

34 Learning Objective 4: Example 2 Step 1: Check the sample size requirement: Is the sample size sufficiently large to use the hypothesis test for a population proportion? Is np 0 >15 and n(1-p 0 ) >15? 54(1/7) = 7.7 and 54(6/7) = 46.3 The first, np 0 is not large enough We will see that the two-sided test is robust when this assumption is not satisfied

35 Learning Objective 4: Example 2 Step 2: Hypotheses H 0: p = 1/7 H a: p ≠ 1/7

36 Learning Objective 4: Example 2 Step 3: Test Statistic

37 Learning Objective 4: Example 2 Step 4: P-value

38 Learning Objective 4: Example 2 Step 5: Conclusion Since the P-value is very small and the sample proportion is greater than 1/7, the evidence strongly suggests that the dogs’ selections are better than random guessing

39 Learning Objective 4: Example 2 Insight: In this study, the subjects were a convenience sample rather than a random sample from some population Also, the dogs were not randomly selected Any inferential predictions are highly tentative. They are valid only to the extent that the patients and the dogs are representative of their populations The predictions become more conclusive if similar results occur in other studies

40 Learning Objective 4: Example 2

41 Learning Objective 5: Summary of P-values for Different Alternative Hypotheses Alternative Hypothesis P-value H a : p > p 0 Right-tail probability H a : p < p 0 Left-tail probability H a : p ≠ p 0 Two-tail probability

42 Learning Objective 6: The Significance Level Tells Us How Strong the Evidence Must Be Sometimes we need to make a decision about whether the data provide sufficient evidence to reject H 0 Before seeing the data, we decide how small the P-value would need to be to reject H 0 This cutoff point is called the significance level

43 Learning Objective 6: The Significance Level Tells Us How Strong the Evidence Must Be

44 Learning Objective 6: Significance Level The significance level is a number such that we reject H 0 if the P-value is less than or equal to that number In practice, the most common significance level is 0.05 When we reject H 0 we say the results are statistically significant

45 Learning Objective 6: Possible Decisions in a Hypothesis Test P-value:Decision about H 0 : ≤ αReject H 0 > αFail to reject H 0

46 Learning Objective 6: Report the P-value Learning the actual P-value is more informative than learning only whether the test is “statistically significant at the 0.05 level” The P-values of 0.01 and are both statistically significant in this sense, but the first P-value provides much stronger evidence against H 0 than the second

47 Learning Objective 6: “Do Not Reject H 0 ” Is Not the Same as Saying “Accept H 0 ” Analogy: Legal trial Null Hypothesis: Defendant is Innocent Alternative Hypothesis: Defendant is Guilty If the jury acquits the defendant, this does not mean that it accepts the defendant’s claim of innocence Innocence is plausible, because guilt has not been established beyond a reasonable doubt

48 Learning Objective 7: One-Sided vs Two-Sided Tests Things to consider in deciding on the alternative hypothesis: The context of the real problem In most research articles, significance tests use two-sided P-values Confidence intervals are two-sided

49 Learning Objective 8: The Binomial Test for Small Samples The test about a proportion assumes normal sampling distributions for and the z-test statistic. It is a large-sample test because the CLT requires that the expected numbers of successes and failures be at least 15. In practice, the large-sample z test still performs quite well in two-sided alternatives even for small samples. Warning: For one-sided tests, when p 0 differs from 0.50, the large-sample test does not work well for small samples

50 Learning Objective 9: Class Exercise 1 In a survey by Media General and the Associated Press, 813 of the 1084 respondents indicated support for a ban on household aerosols. At the 1% significance level, test the claim that more than 70% of the population supports the ban.

51 Learning Objective 9: Class Exercise 2 In a Roper Organization poll of 2000 adults, 1280 have money in regular savings accounts. Use this sample data to test the claim that less than 65% of all adults have money in regular savings accounts. Use a 5% level of significance.

52 Learning Objective 9: Class Exercise 3 According to a Harris Poll, 71% of Americans believe that the overall cost of lawsuits is too high. If a random sample of 500 people results in 74% who hold that belief, test the claim that the actual percentage is 71%. Use a 10% significance level.

53 Chapter 9: Statistical Inference: Significance Tests about Hypotheses Section 9.3: Significance Tests About Means

54 Learning Objectives 1. Steps of a Significance Test about a Population Mean 2. Summary of P-values for Different Alternative Hypotheses 3. Example: Significance Test for a Population Mean 4. Results of Two-Sided Tests and Results of Confidence Intervals Agree 5. What If the Population Does Not Satisfy the Normality Assumption? 6. Regardless of Robustness, Look at the Data

55 Learning Objective 1: Steps of a Significance Test About a Population Mean Step 1: Assumptions The variable is quantitative The data are obtained using randomization The population distribution is approximately normal. This is most crucial when n is small and H a is one- sided.

56 Learning Objective 1: Steps of a Significance Test About a Population Mean Step 2: Hypotheses: The null hypothesis has the form: H 0 : µ = µ 0 The alternative hypothesis has the form: H a : µ > µ 0 (one-sided test) or H a : µ < µ 0 (one-sided test) or H a : µ ≠ µ 0 (two-sided test)

57 Learning Objective 1: Steps of a Significance Test About a Population Mean Step 3: Test Statistic The test statistic measures how far the sample mean falls from the null hypothesis value µ 0, as measured by the number of standard errors between them The test statistic is:

58 Learning Objective 1: Steps of a Significance Test About a Population Mean Step 4: P-value The P-value summarizes the evidence It describes how unusual the data would be if H 0 were true

59 Learning Objective 1: Steps of a Significance Test About a Population Mean Step 5: Conclusion We summarize the test by reporting and interpreting the P-value

60 Learning Objective 2: Summary of P-values for Different Alternative Hypotheses Alternative Hypothesis P-value H a : µ > µ 0 Right-tail probability from t distribution H a : µ < µ 0 Left-tail probability from t distribution H a : µ ≠ µ 0 Two-tail probability from t distribution

61 Learning Objective 3: Example: Mean Weight Change in Anorexic Girls A study compared different psychological therapies for teenage girls suffering from anorexia The variable of interest was each girl’s weight change: ‘weight at the end of the study’ – ‘weight at the beginning of the study’

62 Learning Objective 3: Example: Mean Weight Change in Anorexic Girls One of the therapies was cognitive therapy In this study, 29 girls received the therapeutic treatment The weight changes for the 29 girls had a sample mean of 3.00 pounds and standard deviation of 7.32 pounds

63 Learning Objective 3: Example: Mean Weight Change in Anorexic Girls

64 Learning Objective 3: Example: Mean Weight Change in Anorexic Girls How can we frame this investigation in the context of a significance test that can detect whether the therapy was effective? Null hypothesis: “no effect” Alternative hypothesis: therapy is “effective”

65 Learning Objective 3: Example: Mean Weight Change in Anorexic Girls Step 1: Assumptions The variable (weight change) is quantitative The subjects were a convenience sample, rather than a random sample. The question is whether these girls are a good representation of all girls with anorexia. The population distribution is approximately normal

66 Learning Objective 3: Example: Mean Weight Change in Anorexic Girls Step 2: Hypotheses H 0: µ = 0 H a: µ > 0

67 Learning Objective 3: Example: Mean Weight Change in Anorexic Girls Step 3: Test Statistic

68 Learning Objective 3: Example: Mean Weight Change in Anorexic Girls Step 4: P-value The P-value is the area to the right of t=2.21 for the t sampling distribution with 28 df. This values is If the treatment had no effect, the probability of obtaining a sample this extreme would be 0.018

69 Learning Objective 3: Example: Mean Weight Change in Anorexic Girls Step 5: Conclusion The small P-value of provides considerable evidence against the null hypothesis (the hypothesis that the therapy had no effect)

70 Learning Objective 3: Example: Mean Weight Change in Anorexic Girls “The diet had a statistically significant positive effect on weight (mean change = 3 pounds, n = 29, t = 2.21, P-value = 0.018)” The effect, however, may be small in practical terms 95% CI for µ: (0.2, 5.8) pounds

71 Learning Objective 3: Example: Does Low Carb Diet Work? After 16 weeks on a diet, 41 subjects lost an average of 9.7 kg with a standard deviation of 3.4 kg Calculate the P-value for testing: Ho: μ=0 Ha: μ<0

72 Learning Objective 4: Results of Two-Sided Tests and Results of Confidence Intervals Agree Conclusions about means using two-sided significance tests are consistent with conclusions using confidence intervals If P-value ≤ 0.05 in a two-sided test, a 95% confidence interval does not contain the value specified by the null hypothesis If P-value > 0.05 in a two-sided test, a 95% confidence interval does contain the value specified by the null hypothesis

73 Learning Objective 5: What If the Population Does Not Satisfy the Normality Assumption? For large samples (roughly about 30 or more) this assumption is usually not important The sampling distribution of is approximately normal regardless of the population distribution

74 Learning Objective 5: What If the Population Does Not Satisfy the Normality Assumption? In the case of small samples, we cannot assume that the sampling distribution of is approximately normal Two-sided inferences using the t distribution are robust against violations of the normal population assumption. They still usually work well if the actual population distribution is not normal The test does not work well for a one-sided test with small n when the population distribution is highly skewed

75 Learning Objective 6: Regardless of Robustness, Look at the Data Whether n is small or large, you should look at the data to check for severe skew or for severe outliers In these cases, the sample mean could be a misleading measure

76 Chapter 9: Statistical Inference: Significance Tests about Hypothesis Section 9.4: Decisions and Types of Errors in Significance Tests

77 Learning Objectives 1. Type I and Type II Errors 2. Significance Test Results 3. Type I Errors 4. Type II Errors   and Power

78 Learning Objective 1: Type I and Type II Errors When H 0 is true, a Type I Error occurs when H 0 is rejected When H 0 is false, a Type II Error occurs when H 0 is not rejected As P(Type I Error) goes Down, P(Type II Error) goes Up The two probabilities are inversely related

79 Learning Objective 2: Significance Test Results

80 Learning Objective 3: Decision Errors: Type I If we reject H 0 when in fact H 0 is true, this is a Type I error. If we decide there is a significant relationship in the population (reject the null hypothesis): This is an incorrect decision only if H 0 is true. The probability of this incorrect decision is equal to . If we reject the null hypothesis when it is true and  = 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 incorrectly reject the null hypothesis and conclude significance.

81 Learning Objective 3: P(Type I Error) = Significance Level α Suppose H 0 is true. The probability of rejecting H 0, thereby committing a Type I error, equals the significance level, α, for the test. We can control the probability of a Type I error by our choice of the significance level The more serious the consequences of a Type I error, the smaller α should be

82 Learning Objective 4: Decision Errors: Type II If we fail to reject H 0 when in fact H a is true, this is a Type II error. If we decide not to reject the null hypothesis and thus allow for the plausibility of the null hypothesis We make an incorrect decision only if H a is true. The probability of this incorrect decision is denoted by 

83 The probability that a fixed level  significance test will reject H 0 when a particular alternative value of the parameter is true is called the power of the test against that specific alternative value. Power = 1- . While  gives the probability of wrongly rejecting H 0 when in fact H 0 is true, power gives the probability of correctly rejecting H 0 when in fact H 0 should be rejected (because the value of the parameter is some specific value satisfying the alternative hypothesis) When  is close to  0, the test will find it hard to distinguish between the two (low power); however, when  is far from  0, the test will find it easier to find a difference (high power). Learning Objective 5:  and Power

84 Chapter 9: Statistical Inference: Significance Tests about Hypothesis Section 9.5: Limitations of Significance Tests

85 Learning Objective 1. Statistical Significance vs. Practical Significance 2. Significance Tests Are Less Useful Than Confidence Intervals 3. Misinterpretations of Results of Significance Tests 4. Where Did the Data Come From?

86 Learning Objective 1: Statistical Significance Does Not Mean Practical Significance When we conduct a significance test, its main relevance is studying whether the true parameter value is: Above, or below, the value in H 0 and Sufficiently different from the value in H 0 to be of practical importance

87 Learning Objective 1: What the Significance Test Tells Us The test gives us information about whether the parameter differs from the H 0 value and its direction from that value It does not tell us about the practical importance of the results

88 Learning Objective 1: Statistical Significance vs. Practical Significance When the sample size is very large, tiny deviations from the null hypothesis (with little practical consequence) may be found to be statistically significant. When the sample size is very small, large deviations from the null hypothesis (of great practical importance) might go undetected (statistically insignificant). Statistical significance is not the same thing as practical significance.

89 Learning Objective 1: Statistical Significance vs. Practical Significance A small P-value, such as 0.001, is highly statistically significant, but it does not imply an important finding in any practical sense In particular, whenever the sample size is large, small P-values can occur when the point estimate is near the parameter value in H 0

90 Learning Objective 2: Significance Tests Are Less Useful Than Confidence Intervals A significance test merely indicates whether the particular parameter value in H 0 is plausible When a P-value is small, the significance test indicates that the hypothesized value is not plausible, but it tells us little about which potential parameter values are plausible A Confidence Interval is more informative, because it displays the entire set of believable values

91 Learning Objective 3: Misinterpretations of Results of Significance Tests “Do Not Reject H 0 ” does not mean “Accept H 0 ” A P-value above 0.05 when the significance level is 0.05, does not mean that H 0 is correct A test merely indicates whether a particular parameter value is plausible

92 Learning Objective 3: Misinterpretations of Results of Significance Tests Statistical significance does not mean practical significance A small P-value does not tell us whether the parameter value differs by much in practical terms from the value in H 0 The P-value cannot be interpreted as the probability that H 0 is true

93 Learning Objective 3: Misinterpretations of Results of Significance Tests It is misleading to report results only if they are “statistically significant” Some tests may be statistically significant just by chance True effects may not be as large as initial estimates reported by the media

94 Learning Objective 3: Misinterpretations of Results of Significance Tests If H 0 represents an assumption that people have believed in for years, strong evidence (small P-value) will be needed to persuade them otherwise. If the consequences of rejecting H 0 are great (such as making an expensive or difficult change from one procedure or type of product to another), then strong evidence as to the benefits of the change will be required. Although  = 0.05 is a common cut-off for the P-value, there is no set border between “significant” and “insignificant,” only increasingly strong evidence against H 0 (in favor of H a ) as the P- value gets smaller.

95 When you use statistical inference, you are acting as if your data are a probability sample or come from a randomized experiment. Statistical confidence intervals and tests cannot remedy basic flaws in producing data, such as voluntary response samples or uncontrolled experiments. If the data do not come from a probability sample or a randomized experiment, the conclusions may be open to challenge. To answer the challenge, ask whether the data can be trusted as a basis for the conclusions of the study. Learning Objective 4: Where Did the Data Come From?

96 Chapter 9:Statistical Inference: Significance Tests about Hypothesis Section 9.6: How Likely is a Type II Error

97 Learning Objectives 1. Type II Error 2. Calculating Type II Error 3. Power of a Test

98 Learning Objective 1: Type II Error A Type II error occurs in a hypothesis test when we fail to reject H 0 even though it is actually false To calculate the probability of a Type II error, we must perform separate calculations for various values of the parameter of interest

99 Learning Objective 2: Example 1: Calculating a Type II Error Scientific “test of astrology” experiment: For each of 116 adult volunteers, an astrologer prepared a horoscope based on the positions of the planets and the moon at the moment of the person’s birth Each adult subject also filled out a California Personality Index (CPI) Survey

100 For a given adult, his or her birth data and horoscope were shown to the astrologer together with the results of the personality survey for that adult and for two other adults randomly selected from the group The astrologer was asked which personality chart of the 3 subjects was the correct one for that adult, based on his or her horoscope Learning Objective 2: Example 1: Calculating a Type II Error

astrologers were randomly chosen to take part in the experiment The National Council for Geocosmic Research claimed that the probability of a correct guess on any given trial in the experiment was larger than 1/3, the value for random guessing Learning Objective 2: Example 1: Calculating a Type II Error

102 With random guessing, p = 1/3 The astrologers’ claim: p > 1/3 The hypotheses for this test:  H o : p = 1/3  H a : p > 1/3 The significance level used for the test is 0.05 Learning Objective 2: Example 1: Calculating a Type II Error

103 For what values of the sample proportion can we reject H 0 ? A test statistic of z = has a P-value of So, we reject H 0 for z ≥ and we fail to reject H 0 for z < Learning Objective 2: Example 1: Calculating a Type II Error

104 Find the value of the sample proportion that would give us a z of 1.645: Learning Objective 2: Example 1: Calculating a Type II Error

105 So, we fail to reject H 0 if Suppose that in reality astrologers can make the correct prediction 50% of the time (that is, p = 0.50) In this case, (p = 0.50), we can now calculate the probability of a Type II error Learning Objective 2: Example 1: Calculating a Type II Error

106 We calculate the probability of a sample proportion < assuming that the true proportion is 0.50 Learning Objective 2: Example 1: Calculating a Type II Error

107 The area to the left of in the standard normal table is 0.02 The probability of making a Type II error and failing to reject H 0 : p = 1/3 is only 0.02 in the case for which the true proportion is 0.50 This is only a small chance of making a Type II error Learning Objective 2: Example 1: Calculating a Type II Error

108 Learning Objective 1: Type II Error For a fixed significance level , P(Type II error) decreases as the parameter value moves farther into the H a values and away from the H 0 value as the sample size increases

109 Learning Objective 3: Power of a Test Power = 1 – P(Type II error)=probability of rejecting the null hypothesis when it is false The higher the power, the better In practice, it is ideal for studies to have high power while using a relatively small significance level

110 In this example, the Power of the test when p = 0.50 is: 1 – 0.02 = 0.98 Since, the higher the power the better, a test with power of 0.98 is quite good Learning Objective 3: Power of a Test