Significance Tests About Section 8.2 Significance Tests About Proportions

Example: Are Astrologers’ Predictions Better Than Guessing? 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 Survey

Example: Are Astrologers’ Predictions Better Than Guessing? For a given adult, his or her birth data and horoscope were shown to an 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

Example: Are Astrologers’ Predictions Better Than Guessing? 28 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

Example: Are Astrologers’ Predictions Better Than Guessing? Put this investigation in the context of a significance test by stating null and alternative hypotheses

Example: Are Astrologers’ Predictions Better Than Guessing? With random guessing, p = 1/3 The astrologers’ claim: p > 1/3 The hypotheses for this test: Ho: p = 1/3 Ha: p > 1/3

What Are the 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

What Are the Steps of a Significance Test about a Population Proportion? Step 2: Hypotheses The null hypothesis has the form: Ho: p = po The alternative hypothesis has the form: Ha: p > po (one-sided test) or Ha: p < po (one-sided test) or Ha: p ≠ po (two-sided test)

What Are the 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, po, relative to what we’d expect if Ho were true The test statistic is:

The P-value summarizes the evidence What Are the Steps of a Significance Test about a Population Proportion? Step 4: P-value The P-value summarizes the evidence It describes how unusual the data would be if H0 were true

We summarize the test by reporting and interpreting the P-value What Are the Steps of a Significance Test about a Population Proportion? Step 5: Conclusion We summarize the test by reporting and interpreting the P-value

Example: Are Astrologers’ Predictions Better Than Guessing? Step 1: Assumptions The data is categorical – each prediction falls in the category “correct” or “incorrect” prediction 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

Example: Are Astrologers’ Predictions Better Than Guessing? Step 2: Hypotheses H0: p = 1/3 Ha: p > 1/3

Example: Are Astrologers’ Predictions Better Than Guessing? Step 3: Test Statistic: In the actual experiment, the astrologers were correct with 40 of their 116 predictions (a success rate of 0.345)

Example: Are Astrologers’ Predictions Better Than Guessing? Step 4: P-value The P-value is 0.40

Example: Are Astrologers’ Predictions Better Than Guessing? Step 5: Conclusion The P-value of 0.40 is not especially small It does not provide strong evidence against H0: p = 1/3 There is not strong evidence that astrologers have special predictive powers

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 H0 is true The burden of proof is on Ha

How Do We Interpret the P-value? The approach used in hypotheses testing is called a proof by contradiction To convince ourselves that Ha is true, we must show that data contradict H0 If the P-value is small, the data contradict H0 and support Ha

Two-Sided Significance Tests A two-sided alternative hypothesis has the form Ha: p ≠ p0 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

Example: Dr Dog: Can Dogs Detect Cancer by Smell? Study: investigate whether dogs can be trained to distinguish a patient with bladder cancer by smelling compounds released in the patient’s urine

Example: Dr Dog: Can Dogs Detect Cancer by Smell? 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

Example: Dr Dog: Can Dogs Detect Cancer by Smell? 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)

Example: Dr Dog: Can Dogs Detect Cancer by Smell? Does this study provide strong evidence that the dogs’ predictions were better or worse than with random guessing?

Example: Dr Dog: Can Dogs Detect Cancer by Smell? Step 1: Check the sample size requirement: Is the sample size sufficiently large to use the hypothesis test for a population proportion? Is np0 >15 and n(1-p0) >15? 54(1/7) = 7.7 and 54(6/7) = 46.3 The first, np0 is not large enough We will see that the two-sided test is robust when this assumption is not satisfied

Example: Dr Dog: Can Dogs Detect Cancer by Smell? Step 2: Hypotheses H0: p = 1/7 Ha: p ≠ 1/7

Example: Dr Dog: Can Dogs Detect Cancer by Smell? Step 3: Test Statistic

Example: Dr Dog: Can Dogs Detect Cancer by Smell? Step 4: P-value

Example: Dr Dog: Can Dogs Detect Cancer by Smell? 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

Summary of P-values for Different Alternative Hypotheses Alternative Hypothesis P-value Ha: p > p0 Right-tail probability Ha: p < p0 Left-tail probability Ha: p ≠ p0 Two-tail probability

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 H0 Before seeing the data, we decide how small the P-value would need to be to reject H0 This cutoff point is called the significance level

The Significance Level Tells Us How Strong the Evidence Must Be

Significance Level The significance level is a number such that we reject H0 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 H0 we say the results are statistically significant

Possible Decisions in a Test with Significance Level = 0.05 P-value: Decision about H0: ≤ 0.05 Reject H0 > 0.05 Fail to reject H0

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 0.049 are both statistically significant in this sense, but the first P-value provides much stronger evidence against H0 than the second

“Do Not Reject H0” Is Not the Same as Saying “Accept H0” 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

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

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 the 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 p0 differs from 0.50, the large-sample test does not work well for small samples

Significance Tests about Means Section 8.3 Significance Tests about Means

What Are the 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 Ha is one-sided.

What Are the Steps of a Significance Test about a Population Mean? Step 2: Hypotheses: The null hypothesis has the form: H0: µ = µ0 The alternative hypothesis has the form: Ha: µ > µ0 (one-sided test) or Ha: µ < µ0 (one-sided test) or Ha: µ ≠ µ0 (two-sided test)

What Are the 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 µ0 relative to what we’d expect if H0 were true The test statistic is:

What Are the 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 H0 were true

What Are the Steps of a Significance Test about a Population Mean? Step 5: Conclusion We summarize the test by reporting and interpreting the P-value

Summary of P-values for Different Alternative Hypotheses Alternative Hypothesis P-value Ha: µ > µ0 Right-tail probability Ha: µ < µ0 Left-tail probability Ha: µ ≠ µ0 Two-tail probability

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’

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

Example: Mean Weight Change in Anorexic Girls

Example: Mean Weight Change in Anorexic Girls How can we frame this investigation in the context of a significance test that can detect a positive or negative effect of the therapy? Null hypothesis: “no effect” Alternative hypothesis: therapy has “some effect”

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

Example: Mean Weight Change in Anorexic Girls Step 2: Hypotheses H0: µ = 0 Ha: µ ≠ 0

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

Example: Mean Weight Change in Anorexic Girls Step 4: P-value Minitab Output Test of mu = 0 vs not = 0 Variable N Mean StDev SE Mean wt_chg 29 3.000 7.3204 1.3594 CI 95% CI T P (0.21546, 5.78454) 2.21 0.036

Example: Mean Weight Change in Anorexic Girls Step 5: Conclusion The small P-value of 0.036 provides considerable evidence against the null hypothesis (the hypothesis that the therapy had no effect)

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 = 0.04)” The effect, however, may be small in practical terms 95% CI for µ: (0.2, 5.8) pounds

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 H0 value If P-value > 0.05 in a two-sided test, a 95% confidence interval does contain the H0 value

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 x is approximately normal regardless of the population distribution

What If the Population Does Not Satisfy the Normality Assumption In the case of small samples, we cannot assume that the sampling distribution of x 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

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