Presentation on theme: "Confidence Intervals and Hypothesis Tests"— Presentation transcript:
1 Confidence Intervals and Hypothesis Tests Section 4.5Confidence Intervals and Hypothesis Tests
2 Bootstrap and Randomization Distributions Bootstrap DistributionRandomization DistributionEstimates the distribution of sample statisticsEstimates the distribution of sample statistics, if H0 trueCentered around the observed sample statisticCentered around the null hypothesized valueSimulate sampling from the population by resampling from the original sampleSimulate samples assuming H0 were trueBig difference: a randomization distribution assumes H0 is true, while a bootstrap distribution does not
3 Which Distribution?Let be the average amount of sleep college students get per night. Data was collected on a sample of students, and for this sample hours.A bootstrap distribution is generated to create a confidence interval for , and a randomization distribution is generated to see if the data provide evidence that > 7.Which distribution below is the bootstrap distribution?(a) is centered around the sample statistic, 6.7
4 Which Distribution?Intro stat students are surveyed, and we find that 152 out of 218 are female. Let p be the proportion of intro stat students at that university who are female.A bootstrap distribution is generated for a confidence interval for p, and a randomization distribution is generated to see if the data provide evidence that p > 1/2.Which distribution is the randomization distribution?(a) is centered around the null value, 1/2
5 Body TemperatureWe created a bootstrap distribution for average body temperature by resampling with replacement from the original sample (
6 Body TemperatureWe also created a randomization distribution to see if average body temperature differs from 98.6F by adding 0.34 to every value to make the null true, and then resampling with replacement from this modified sample:
7 Body TemperatureThese two distributions are identical (up to random variation from simulation to simulation) except for the centerThe bootstrap distribution is centered around the sample statistic, 98.26, while the randomization distribution is centered around the null hypothesized value, 98.6The randomization distribution is equivalent to the bootstrap distribution, but shifted over
8 Body Temperature Bootstrap Distribution Randomization Distribution 98.2698.6Randomization DistributionH0: = 98.6Ha: ≠ 98.6Talk about the fact that the null hypothesized value is in the extremes of the bootstrap distribution, so the sample statistic is in the extremes of the randomization distribution
9 Body Temperature Bootstrap Distribution Randomization Distribution 98.2698.4Randomization DistributionH0: = 98.4Ha: ≠ 98.4Talk about the fact that the null hypothesized value is not in the extremes of the bootstrap distribution, so the sample statistic is not in the extremes of the randomization distribution
10 Intervals and TestsIf a 95% CI contains the parameter in H0, then a two-tailed test should not reject H0 at a 5% significance level.If a 95% CI misses the parameter in H0, then a two-tailed test should reject H0 at a 5% significance level.
11 Intervals and TestsA confidence interval represents plausible values for the population parameterIf the null hypothesized value IS NOT within the CI, it is not plausible and should be rejectedIf the null hypothesized value IS within the CI, it is plausible and should not be rejected
12 Body TemperaturesUsing bootstrapping, we found a 95% confidence interval for the mean body temperature to be (98.05, 98.47)This does not contain 98.6, so at α = 0.05 we would reject H0 for the hypothesesH0 : = 98.6Ha : ≠ 98.6
13 Both Father and Mother“Does a child need both a father and a mother to grow up happily?”Let p be the proportion of adults aged in who say yes. A 95% CI for p is (0.487, 0.573).Testing H0: p = 0.5 vs Ha: p ≠ 0.5 with α = 0.05, do we reject H0 or not reject H0?Do not reject H0 0.5 is within the CI, so is a plausible value for p.
14 Both Father and Mother“Does a child need both a father and a mother to grow up happily?”Let p be the proportion of adults aged in who say yes. A 95% CI for p is (0.533, 0.607).Testing H0: p = 0.5 vs Ha: p ≠ 0.5 with α = 0.05, we reject H0 or do not reject H0?Reject H0 0.5 is not within the CI, so is not a plausible value for p.
15 Intervals and TestsConfidence intervals are most useful when you want to estimate population parametersHypothesis tests and p-values are most useful when you want to test hypotheses about population parametersConfidence intervals give you a range of plausible values; p-values quantify the strength of evidence against the null hypothesis
16 Interval, Test, or Neither? Are the following questions best assessed using a confidence interval, a hypothesis test, or is statistical inference not relevant?Do a majority of adults riding a bicycle wear a helmet?On average, how much more do adults who played sports in high school exercise than adults who did not play sports in high school?On average, were the 23 players on the Canadian Olympic hockey team older than the 23 players on the 2010 US Olympic hockey team?test; interval; inference not relevant
17 Cautions About Significance With small sample sizes, even large differences or effects may not be significantWith large sample sizes, even a very small difference or effect can be significantA statistically significant result is not always practically significant, especially with large sample sizes
18 Statistical vs Practical Significance Example: Suppose a weight loss program recruits 10,000 people for a randomized experiment.A difference in average weight loss of only 0.5 lbs could be found to be statistically significantSuppose the experiment lasted for a year. Is a loss of ½ a pound practically significant?
19 Diet and Sex of BabyAre certain foods in your diet associated with whether or not you conceive a boy or a girl? To study this, researchers asked women about their eating habits, including asking whether or not they ate 133 different foods regularly. For each of the 133 foods studied, a hypothesis test was conducted for a difference between mothers who conceived boys and girls in the proportion who consume each food.
20 Hypothesis Tests State the null and alternative hypotheses If there are NO differences (all null hypotheses are true), about how many significant differences would be found using α = 0.05?A significant difference was found for breakfast cereal (mothers of boys eat more), prompting the headline “Breakfast Cereal Boosts Chances of Conceiving Boys”. How might you explain this?pb: proportion of mothers who have boys that consume the food regularlypg: proportion of mothers who have girls that consume the food regularlyH0: pb = pgHa: pb ≠ pg133 0.05 = 6.65Random chance; several tests (about 6 or 7) are going to be significant, even if no differences exist
21 Multiple TestingWhen multiple hypothesis tests are conducted, the chance that at least one test incorrectly rejects a true null hypothesis increases with the number of tests. If the null hypotheses are all true, α of the tests will yield statistically significant results just by random chance.
22 Multiple ComparisonsConsider a topic that is being investigated by research teams all over the world Using α = 0.05, 5% of teams are going to find something significant, even if the null hypothesis is true
23 Multiple ComparisonsConsider a research team/company doing many hypothesis testsUsing α = 0.05, 5% of tests are going to be significant, even if the null hypotheses are all true
24 Multiple Comparisons This is a serious problem The most important thing is to be aware of this issue, and not to trust claims that are obviously one of many tests (unless they specifically mention an adjustment for multiple testing)There are ways to account for this (e.g. Bonferroni’s Correction), but these are beyond the scope of this class
25 Publication BiasPublication bias refers to the fact that usually only the significant results get publishedThe one study that turns out significant gets published, and no one knows about all the insignificant resultsThis combined with the problem of multiple comparisons, can yield very misleading results
26 Jelly Beans Cause Acne! http://xkcd.com/882/ Consider having your students act this out in class, each reading aloud a different part. it’s very fun!
30 SummaryIf a null hypothesized value lies inside a 95% CI, a two-tailed test using α = 0.05 would not reject H0If a null hypothesized value lies outside a 95% CI, a two-tailed test using α = 0.05 would reject H0Statistical significance is not always the same as practical significanceUsing α = 0.05, 5% of all hypothesis tests will lead to rejecting the null, even if all the null hypotheses are true