1. By Joanna Charteris

2.  posing a comparison investigative question using a given multivariate data set  selecting and using appropriate displays and summary statistics  discussing sample distributions  discussing sampling variability, including the variability of estimates  making an appropriate formal statistical inference  communicating findings in a conclusion.

3.  Achieved - Use statistical methods to make a formal inference involves showing evidence of using each component of the statistical enquiry cycle.  Merit - Use statistical methods to make a formal inference, with justification involves linking components of the statistical enquiry cycle to the context, and referring to evidence such as sample statistics, data values, or features of visual displays in support of statements made.  Excellence - Use statistical methods to make a formal inference, with statistical insight involves integrating statistical and contextual knowledge throughout the statistical enquiry cycle, and may include reflecting about the process; considering other relevant explanations.

6. The mean shower length for females is 25min and the mean shower length for males is 30min. Can we make the call that males spend longer in the shower?? Obviously its not feasible to collect data on all females and males, therefore the mean is based on a sample. What would happen if we took a different sample?

7. To answer our question about what is happening back in the population, we need to understand that we have only calculated estimates based on a sample. If we had a different sample our estimates WILL CHANGE!! If that sample was biased in any way, then it is not a true reflection of what is happening back in the population therefore, our investigation is not valid.

8. Males Females Number of messages sent a day The median of females is bigger, therefore can we say females send more messages a day than males? NO!!! This is based on a sample and is only an estimate. Really we need more information to understand what is happening back in the population.

9. Last year (in MAT202) we used an informal confidence interval to make calls about a population. This allowed for sampling variability and is a way of making valid, accurate conclusions about a population. Remember an informal confidence interval is a range where the true population mean/median is likely to lie between. Males Females Number of messages sent a day This “bar” is the informal confidence interval. This is the range where the true population median is likely to be. If the overlap then, we cant conclude that there is a difference between males and females.

10. At level 3 we use a similar idea. The ICI is just a calculation – so how accurate is it? A better way would be to go out a collect 100 different samples and then find the median of all the sample medians. WHOA! – is this feasible??? This is where the idea of bootstrapping comes in.

11. https://www.youtube.com/watch?v=3Y_Ps4ETwo0 Ponyland is a mystical land, home to all kinds of magical creatures. The Little Ponies make their home in Paradise Estate, living a peaceful life filled with song and games. However, not all of the creatures of Ponyland are so peaceful, and the Ponies often find themselves having to fight for survival against witches, trolls, goblins and all the other beasts that would love to see the Little Ponies destroyed, enslaved or otherwise harmed. [1] [1] https://en.wikipedia.org/wiki/My_Little_Pony_(TV_series)

12. The mean height of Little Ponies at Paradise Estate is 150mm with a standard deviation of 5mm. Sketch a possible height distribution for the population of Little Ponies at Paradise Estate. Remember to give an indication of scale.

14. How tall is your doozer? Casio Graphics calculator: RandNorm#(5, 150) Excel (normsinv(rand())*5+150 TI 84+ calculator: randnorm(150,5) Draw your doozer Tell your neighbor about your doozer

15. I wonder what is the mean height of the Little Ponies at Paradise Estate?  From Level 6 (Year 11) we use means/medians to estimate the population mean; however we know there is too much uncertainty  From Level 7 (Year 12) we use an informal confidence interval to estimate the range where the population mean will lie.  Level 8 (Year 13); you will learn a new analysis tool called Bootstrapping

16. 1. Find the mean of your sample first – write it down 2. Shuffle your Ponies 3. Select 1 and record their height in excel 4. Put that Pony back and re-shuffle 5. Select another Pony 6. Repeat process until you have recorded 10 Pony heights THIS IS YOUR SAMPLE OF 10 Using excel (=average) find the mean of your sample Plot your mean on the board

17. Using iNZight to re-sample  Start iNZight and select the Bootstrap Confidence Interval Construction VIT module.  Import the Pony 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.

19.  Repeat this 3 times (do get another bootstrap make sure you click on record my choices)  Draw each interval by a line on the board.

20. Using iNZight to check how well this method works  Start iNZight and select the Confidence interval coverage VIT module (or select FILE and VIT modules).  Import the Pony 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. Just remember: You will rarely have data on the whole population! This is just a teaching tool to show you how it works!

22. What’s the impact on our confidence intervals? Complete activity

24. Basic facts One thing I didn’t know One thing I found interesting As you may know the link between autism and vaccines has a long and contentious history. Use this topic to do some research into this area. The table below may help you summarise your findings. Come up with AT LEAST two different questions I DO NOT want you to spend much time on this Autism and Vaccines

25. VARIABLE being examined GROUPS being compared POPULATION inferences are being made about STATISTIC being estimated Nightmare Moon is planning to attack her sister as she wanted to lower the moon. The princess wants to fit all the Pegasus Ponies 18years and over with an army uniform. They are unsure if they should make wing guards especially for Females as this will take more time. The Princess has employed you to investigate this problem.

26. VARIABLE being examined Wing length cm GROUPS being compared Male/Females POPULATION inferences are being made about Pegasus Ponies 18 or over STATISTIC being estimated Difference of means Nightmare Moon is planning to attack her sister as she wanted to lower the moon. The princess wants to fit all the Pegasus Ponies 18years and over with an army uniform. They are unsure if they should make wing guards especially for Females as this will take more time. The Princess has employed you to investigate this problem. Complete “Comparison” activity in student resources

27. VARIABLE being examined Wing length cm GROUPS being compared Male/Females POPULATION inferences are being made about Pegasus Ponies 18 or over STATISTIC being estimated Difference of means Nightmare Moon is planning to attack her sister as she wanted to lower the moon. The princess wants to fit all the Pegasus Ponies 18years and over with an army uniform. They are unsure if they should make wing guards especially for Females as this will take more time. The Princess has employed you to investigate this problem. Complete “Comparison” activity in student resources I wonder what the difference is between the mean wing length of Male Pegasus Ponies 18 years or over and Female Pegasus Ponies that are 18 years of age or over

30. SUCCOS S SPREAD Discuss the Inter Quartile Range (IQR) – which is UQ – LQ This is the spread of the middle 50% U UNUSUAL FEATURES This is usually seen by looking at the raw data (dot plot) OR a long whisker C CLUSTERSWhere does most of the data lie between OR any groupings? C CENTRE Compare the middle 50% of the data and which is higher up the scale O OVERLAPIs there a visible overlap of the boxes? S SHAPE What does the shape of the distribution look like? For example, Normally distributed, skewed, uniform, bimodal, irregular.

32. To describe the shape, you need to look at the dot plot!

33. This is ANOTHER acronym to help you to write about features/succos. Obvious Specific Evidence OR Example Meaningful

34. Open run mode Import data Chose your Variable 1 (has to be numerical) Subset by your two groups Import ‘Student Data’ and draw a comparison B & W for the head perimeter between males and females. Get summary Statistics ** Data is based on Year 11 students at Blah College

35. FemaleMale S SPREADAD U UNUSUAL FEATURES C CLUSTERS C CENTRE O OVERLAP S SHAPE

36. FemaleMale SPREAD: Compare the IQR (middle 50% spread) Female IQR = 58 – 55 = 3 Male IQR = 58 – 53.75 = 4.25 The middle 50% of head circumferences belonging to the male year 11 students at Blah College are more spread out than the middle 50% of head circumferences of female Year 11 students at Blah College. This is shown by the male head circumference IQR range being larger by 1.25. This could be because … (possible reason why)

37. FemaleMale Unusual features/value: There is one unusually small head circumference for year 11 males at Blah College at 46cm whereas there are no unusual head circumferences for females at Blah College. This could be because … (possible reason why)

38. FemaleMale Clusters: Most of the head circumferences for Year 11 females at Blah College are between 53cm and 58cm whereas most of the head circumferences for the Year 11 males at Blah College are between 54cm and 58cm. There also seems to be two groupings of Year 11 female students with a head circumference of 57cm and 55cm, whereas the male year 11 students seem to be more scattered with no clusters. This could be because … (possible reason why)

39. FemaleMale Centre: Expectation is to compare the middle 50% Female middle 50% = 58 and 55cm median = 57cm Male middle 50% = 58 and 53.75cm median = 55cm The median head circumference for year 11 female students at Blah College is 2cm bigger than the male Year 11 students at Blah College. The middle 50% of year 11 female students at Blah College is between 55 and 58cm, which is approximately the same as the year 11 male students at Blah College. For example the middle 50% of students have roughly the same head circumference no matter if you were male or female. This could be because … (possible reason why)

40. FemaleMale Overlap: Does the boxes (middle 50%) overlap?? Female middle 50% = 58 and 55cm Male middle 50% = 58 and 53.75cm There is significant overlapping of the middle 50% between male and female year 11 students at Blah College which suggests that we may not be able to make a call whether there is a difference in head circumferences between male and female. This could be because … (possible reason why)

41. FemaleMale Shape: Not really enough information to identify the shape

42. I wonder what is the difference between the mean wing length of Male Pegasus18 years or over and Female Pegasus Ponies that are 18 years of age or over Draw a comparison box and whisker graphs on the wing length of Pegasus Ponies at Paradise Estate Describe any features.

44. Comment on the sample distribution for your TWO investigation questions Heights Spike Copy and paste ANY relevant graphs and/or statistics you have used. Describe the features

47. I am fairly confident that there is a difference between the wing length of female and male Pegasus Ponies that are 18 years or over. I can make the call that Males have long wings than females as the bootstrap values are both positive. I can also say that Male Pegasus Ponies 18 years and over have somewhere between 1.534cm and 3.595cm longer wings than females. I wonder if there are any differences between the mean wing length of Male and Female Pegasus Ponies that are 18 years of age or over

48. Answer both of your comparison questions 1. Open iNZight in bootstrap VIT mode 2. Inport appropriate data 3. Show bootstrap distibution 4. Calculate confidence interval 5. Write a inference. Remember We want to create a bootstrap confidence interval for the difference between median heights of female ponies and median heights of male ponies. We want to create a bootstrap confidence interval for the difference in median heights between the ponies chased by Spike and the ponies not chased by Spike.

49. Complete this sheet in student resources

51. Make a formal statistical inference. Conclude your investigation, reflecting on your hypothesis and justifying your formal inference This may include: -Discussing sampling variability, including the variability of estimates. -Reflecting on the process you have used to make the formal inference

52. I wonder if there are any differences between the mean wing lengths of Male and Female Pegasus Ponies that are 18 years of age or over When looking at the sample variation between male and females, females wing lengths are a lot more spread out than males. However when you compare just the middle 50% spread they only have a difference of 0.65cm which is very small. This leads me to believe that if I had a different sample the spread could potentially be different where there may not be as many female Pegasus Ponies with short wings. If this was the case, this would push up the mean, but may have little effect on the median. Through my research about Pegasus Ponies wings I have learnt that female Pegasus Ponies have a different shape of wing as they are narrower, so looking just at the length of the wing may not be enough to make a recommendation about whether to make special female army wing guards. When looking at the SD and the bootstrap distribution there is not much variation between the difference on means. The interval is also significantly more than 0 which gives me confidence that there is indeed a difference in wing lengths. Based on my investigation and the sample that I was given, I would conclude that there is a difference between male and female wing lengths for all Pegasus Ponies that are 18 years and over. I therefore make the recommendation that they should be making special wing guards for females. Copy and paste question into your conclusion

53. Because our Ponies are fictional we are not going to write a conclusions based on this.

56. Research http://nzta.govt.nz/about/advertising/drink-driving/legend.html