Ideas for Practicals in Statistics and Use of Technology

Transcription

1 Ideas for Practicals in Statistics and Use of Technology Mark Kent, Head of Mathematics and Computing/ICT Faculty The Sixth Form College Solihull There follows a list of ideas for statistics practicals in the classroom. I ve used them all and found them to work well. Feel free to adapt to your circumstances and students. Most importantly, use them as a starting point if they help, to feed your own creativity in the classroom. 1. Binomial Hypothesis Testing to illustrate the critical region in such a test can be difficult for students to grasp. However, AUTOGRAPH provides a quick way to do this. A possible example lesson is this: Talk about research which shows an intake of sugar helps student concentration on higher level (Mathematical) thinking for a short period of time (around 30 minutes). Read out 10 random numbers and then ask students to write them down in the order read out. Read out the answers and get to mark themselves, obtaining a score out of 10. Issue each student with a piece of chocolate which is sugar free (DO NOT TELL THEM THIS!) and ask them to consume it. (iii) Continue the lesson for a further ten minutes, setting the scene for the test, describing hypotheses etc (iv) After 10 minutes repeat above. (v) Open up AUTOGRAPH and establish what the hypotheses would be i.e. H 0 : p = 0.5, H 1 : p > 0.5 (or express in terms of median improvement, giving a sign test) (vi) Click on the sigma symbol (top left) to open up a new statistics page, then click on Data and Enter Probability Distribution select Binomial and choose p=0.5, n = number of students whose results altered between and (iv). You will also need to edit axes AXES then EDIT AXES and change y to between 0 and 0.3, x to between 0 and the number in the sample. (vi) Now select the distribution by clicking on one of the lines in the diagram so that it turns black and choose OBJECT then PROBABILITY CALCULATIONS then CUMULATIVE choose a number. A lozenge appears and by clicking on VIEW then STAUS BOX the cumulative probability is displayed. Some discussion is then possible about the probability necessary to reject H 0. Once decided the test may proceed and usually you end up rejecting H 0, saying thus that the chocolate has a statistically significant effect (not due to random chance) on concentration/memory for year-old students. (vii) Confess that the chocolate used was sugar free and thus we have demonstrated the statistically significant effect of a placebo. Discuss what

2 we would expect to be the effect of sugared chocolate and move on to discussion of drug trials. 2. Mobile Phones - students check time to text college on mobiles with left and right hands in pairs. Data is collected for class. Analysis could be comparative box plot use or hypothesis test (t) on the differences between hands (matched pairs design) 3. Quality Control investigate quality in tubes of smarties. Students count number of smarties which are: deformed/imperfect; red, orange etc. Weigh smarties in a tube. This can be used as a tool to practice control charts (A2 Statistics topic), hypothesis testing (is the median number of defectives greater than say 2? Use a Wilcoxon or sign test), or just basic descriptive statistics. 4. Taste Tests a classic experiment using Binomial Distribution: Students record number of correct choices of say diet coke versus non diet from a selection of 4 cups, three of which have full-sugar coke in. Participants are blindfolded and hold nose whilst drink. B(4,0.25) can be used to predict probability of each number of successes from 0 to 4. Multiplying these probabilities by sample size (number of participants) gives expected frequencies which can be compared with observed and fit checked. Alternatively, perform a Binomial Hypothesis test (AUTOGRAPH can help here) to see whether an average student can tell the difference between the two types of coke Can be repeated using different brands of biscuit, chocolate, coffee etc. 5. Chocolate and Poverty use a bar(s) of Cadbury s Dairy Milk Chocolate and share out squares unevenly around class. When students complain check on mean (fix it so it s say 2 per pupil) and bring in a discussion on best form of average. Now give say 6 extra squares to the child with most average goes up! Lead in to discussion on problems of giving aid to third world countries. Finally, get children to redistribute squares so median, then mode is 2. This lesson works very well, especially at the end of the day when the children/students are a little restive. 6. Choc Chip Cookies can use to introduce/develop paired and unpaired nonparametric tests. Paired Design Each student rates the taste of two different brands of cookie. A two-sample sign test is then carried out. Unpaired Design number of choc chips are counted for a series of cookies of each brand. Data analysed using a two-sample Wilcoxon Test.

3 7. Reactions and Rulers students hold rulers with left and right hands between thumb and fore-finger and release then catch. Do three times with each time, recording distance dropped after third attempt. Plot a scatter diagram of left against right usually highly correlated students find correlation (pmcc). 8. Parental IQ students are told that the average IQ score is 100. They are then asked to estimate both their father s and mother s IQ. The data is recorded on MINITAB or EXCEL, including whether the guesser is male or female. Multiple analyses are then possible, for example: Males tend to score fathers higher than mothers on average, whilst girls tend to be evenly scored for each parent (again on average) Correlation between two sets of scores usually reveals a high positive correlation use Spearman s Rank. 9. Parent Height student and parental (same sex parent) heights are recorded and a regression line found. Can be used to predict height of a particular child. 10. Smoker s Fitness a short, sharp piece of exercise is carried out (press-ups, running on the spot, running up a flight of stairs etc. be careful about health and safety issues here!). Pulse rate before and after is measured. The question being asked is: how fit are smokers? The data is collected, including the gender and whether they smoke. Comparative box plots of the increase in pulse rates will reveal the differences between smokers and nonsmokers. Can also compare genders. Use of a Wilcoxon two sample test also possible, as is a correlation and scatter-diagram of before and after pulse rates. 11. Newspapers used in Poisson and Binomial Distributions Collect a pile of newspapers e.g. Daily Mail, Guardian. Distribute papers and ask students to record in first five pages number of: scandals, feel good/happy stories, number of pictures. Find mean and variance will be roughly equal. Now check fit of Poisson distribution using calculated mean for one of the above categories. Repeat the above but count the number of pages in the first five which have a happy/feel good story on, or a scandal, misprint etc. Check fit of Binomial. Check also mean and use to estimate p. Find theoretical variance and compare with sample variance to further check realism of model. 12. Mars Bars and Memory a good first lesson. The start is the same as 1 above, but using a mars bar (mini) instead. The lesson progresses, but every 5-10 minutes the number of digits correctly remembered out of 10 is recorded on a MINITAB spreadsheet (or EXCEL). At the end of the lesson, use MINITAB to display comparative box plots. It is usually noticeable that memory scores peak after around 20 minutes. This lesson proves very memorable to students if used right at the start of the course.

4 13. Dicey Statistics give each student a die and ask what the most likely number of throws is, up to and including the first six. Each now throws until get a six tally the results on the board. Repeat for bigger sample size. Results usually confound expectations what people feel to be instinctively the most likely outcome, is usually the average or expected value instead. If you ask what the hardest score to get on the die is, most will say six despite the fact that they are all equally likely. This is of course because six is the highest numerical value, and shows a misunderstanding of probability and equal likelihood. 14. A trip to the grave find the nearest large church graveyard and divide the area into distinct sections. Assign groups of students to each and using dice and coins, starting at a random point in each area, students move (left, right forward or back, then a given number of paces). The nearest grave is selected and data copied down e.g. age at death, date of burial, occupation (if there), married or single, gender, size of grave stone etc. Data is then pooled back at school and can then form the basis of class work, projects and just be available as a source of real data as the year progresses. The data can also be used for the GCSE Statistics coursework.

5 AUTOGRAPH USEAGE A FEW IDEAS 1. Distributions-Binomial, Poisson, Normal Open a new statistics page (sigma icon top left of tool bar) Click on DATA-ENTER PROBABILITY DISTRIBUTION choose distribution and parameters. (iii) It is possible to animate the distribution click on the lightening icon and adjust the parameters manually or automatically. 2. Descriptive statistics Open a new statistics page. Click on DATA-ENTER GROUPED DATA Class intervals-equal intervals, integer data or manual Frequencies-either manual or calculated from raw data Edit raw data dialogue box-manual entry import or cut/paste from a spreadsheet create sample from a prob. Distribution Click on OBJECT and select diagram type. 3. Bivariate data Open a 2D graph page Click on DATA- ENTER XY DATA SET (iii) Data can be entered directly or cut/paste from 2 spreadsheet columns or created from a set of selected cursors (using OBJECT) or created from a selected graph. (iv) Click on OBJECT-can add CENTROID (x,y) REGRESSION LINE Results displayed in the RESULTS BOX (icon on tool bar or VIEW- RESULTS BOX) Illustrating residuals Open a 2D graph page-click on equal aspect Create a data set using cursors OBJECT - SELECT ALL POINTS - CENTROID (iii) Attach cursor to centroid-add another cursor and select both. (iv) Click on OBJECT STRAIGHT LINE (v) Select straight line and data set. Click on OBJECT-Y on X RESIDUAL (vi) Drag 2 nd cursor about to vary line and minimize sum of R 2