Inference

1. Inference AS 91582

2. Every time you over-indulge, your life shortens – expert • 11:40 AM Wednesday Dec 19, 2012 • http://www.nzherald.co.nz/lifestyle/news/article.cfm?c_id=6&objectid=10855064

3. Every time you over-indulge, your life shortens - expert • It's the season for eating, drinking and being merry, but experts warn that every time you over-indulge you could be cutting hours off your life. • A new report, published in the British Medical Journal, claims activities like having a couple of drinks, smoking, eating red meat and sitting in front of the TV can cut at least 30 minutes off a person's life for every day that do it.

4. On the flip side, sticking to one alcoholic beverage, eating plenty of fruit and vegetables and working up a sweat can add a couple of hours on to your life, Medical Dailyreports. • Professor David Spiegelhalter, a statistician from the University of Cambridge, figured out the impact of different activities on a person's lifespan by using a concept of accelerated or decelerated ageing.

5. Prof Spiegelhalter hopes expressing activities in microlives like this will help people make better lifestyle decisions. • "Each day of smoking 20 cigarettes is as if you are rushing towards your death," he said.

6. "Of course, evaluation studies would be needed to quantify any effect on behaviour, but one does not need a study to conclude that people do not generally like the idea of getting older faster.”

7. Maximise your chance of living to 100 – Tails you win: the science of chance www.magsmaths.com

8. Meet the Doozers • www.magsmaths.com

9. Height distribution of Doozers at Fraggle Rock • The mean height of Doozers at Fraggle Rock is 150mm with a standard deviation of 5mm

10. Height distribution of Doozers at Fraggle Rock • The mean height of Doozers at Fraggle Rock is 150mm with a standard deviation of 5mm • Sketch a possible height distribution of Doozers at Fraggle Rock. Remember to give an indication of scale.

11. Height distribution of the population of Doozers at Fraggle Rock

12. Problem I would like to get an estimate of the mean height of Doozers in Fraggle Rock.

13. Bootstrapping Outline of the method: • Re-sample with replacement from our original sample. • Create a re-sample that is the same size as our original random sample • Calculate the mean (or statistic of interest) for the re-sample

16. INZight Bootstrapping using iNZight

17. iNZight • Start iNZight and select the Sampling Variation VIT module (or select FILE and VIT modules). • Import the Doozer height population file. • Drag “Height” down to the variable 1 box, and then click the Analyse tab. • The default quantity is mean. Do NOT change this. Change Sample Size to 10, then click on Record my choices. • Play. Check you know what each selection does, and how it relates to sampling from the population.

18. 1000 samples of size 10 from the entire population.

19. We can see that samples of size 10 form a Normal Distribution as the population was Normally Distributed.

20. Bootstrap Intervals • Using iNZight to create a bootstrap confidence interval • Start iNZight and select the Bootstrap Confidence Interval Construction VIT module. • Import the Doozer sample session 1 file. • Drag Height down to the variable 1 box, and then click the Analyse tab. • The default quantity is “mean”. Do NOT change this, just click on “Record my choices” • Play, and replicate what you have just done by hand. Check you know what each selection does. • To finish, copy and paste the Bootstrap distribution of re-sample means into a word document.

21. Distribution from the sample

22. iNZight • Start iNZight and select the Confidence interval coverage VIT module (or select FILE and VIT modules). • Import the Doozer height population file. • Drag “Height” down to the variable 1 box, and then click the Analyse tab. • The default quantity is mean. Do NOT change this. Change the CI Method to bootstrap: percentile and the Sample Size to 10, then click on Record my choices. • Play. Check you know what each selection does, and how it relates to the bootstrap confidence intervals.

23. How well does the method work We can use iNZight to check how well the Bootstrap method works, by repeating the process many times taking different random samples from a known population.

24. Key Points The distribution of re-sample means (the bootstrap distribution) is similar to the distribution of means from repeated random sampling. Therefore we can use the bootstrap distribution to model the sampling variability in our data, and base our confidence interval on this bootstrap distribution.

25. NOTE The method doesn’t rely on having a Normal distribution and it works for small numbers.

26. Confidence Interval coverage

27. Increasing the sample size – what’s the impact on our confidence intervals? • We will use the Doozer height data to investigate this. Each file contains 10 different random samples from the population of Doozers at Fraggle Rock.

29. What seems to happen to the widths of the bootstrap confidence intervals when you increase the sample size? Consider the following: • How much does the width change? • How could you describe the change? • Can you describe how the width is related to the number of Doozers in the samples?