A



Worksheets for Workshop Sat 1 April

1 Vocabulary

2 Beetle – an adaptation of an activity in ‘Chance and Data’ Volume 1

3 Chocolate Chippie Biscuits –

adaptation of an activity in ‘Chance and Data’ Volume 1

4 Investigating confidence intervals

5 Analysing Bivariate Data

6 Sources of inspiration – books and websites

7 Dataset descriptor

PROBABILITY -

TERMS AND DEFINITIONS

Match the terms in the left-hand column with the correct definition or formula in the right-hand column

|Random Experiment |A variable whose outcome is determined by the outcomes of the |

| |experiment. Denoted by X |

|Trial |P(A) |

| | |

|Outcome |Gives the likelihood of different values. Often expressed |

| |using a table or formula |

|Sample space |Process, a result of which depends on chance |

|Event |Usually takes whole number values only |

|Random Variable |Probability of 0 |

| | |

|Discrete Random Variable |Between 0 and 1 inclusive |

| | |

|Continuous Random Variable |Number of favourable outcomes |

| |Total number of trials |

|Impossible Event |One performance of the experiment |

|Certain Event |A subset of the Sample Space |

| | |

|Probability of event A occurring |Can take any real number value in a certain range |

|Range of values for a probability |Number of elements in the event |

| |No. of elements in the Sample Space |

|Experimental Probability (Long-run frequency) |A result of the experiment |

| | |

|Theoretical Probability (with equally likely outcomes) |Probability of 1 |

| | |

|Probability function |Set of all possible outcomes |

| | |

Beetle

1. Set up a tally chart and frequency table to record dice throws on.

2 We will play Beetle with the following rules

• record all throws of the die on your tally chart

• you must draw a body first

• a head is needed before eyes, feelers or mouth can be drawn

• body parts are drawn from the following throws

6. body

5. head

4. mouth

3 leg - need six

2 eye - need two

1. feeler - need two

• the first person to complete a beetle is the winner!

2. When you have completed your beetle, add up how many of each number you got and calculate the relative frequency of each number.

3. Draw a bar graph showing the frequency of each number

4. Record your results on the class chart.

What changes when we look at a larger sample?

Draw a bar graph showing the relative frequency of each number for the class.

How do these compare to the theoretical probabilities of each number?

HOW MANY CHOCOLATE CHIPS MAKES A GOOD CHOCOLATE CHIPPIE BISCUIT?

[pic]

[pic]

1 How many chocolate chips are there in a bought chocolate chippie biscuit? Eat and count!

2 How could we simulate stirring 100chocolate chips into 10 cookies?

Try it – what happens?

3 How many chocolate chips does each biscuit need? How many chocolate chips would we need for 10 biscuits to ensure they all (most?) get this many?

4 How could we model this using the binomial distribution?

Does this change your answer to 3?

[pic][pic][pic][pic]

Investigating confidence intervals

Complete the probability distribution table for X where X 1-digit random numbers

|x |0 |

|50 |7.0 |

|50 |7.5 |

|100 |12.8 |

|200 |24.0 |

|400 |47.0 |

|800 |93.0 |

|1200 |138.0 |

|1600 |183.0 |

|2000 |230.0 |

|2000 |226.0 |

1. Plotted on the scatter diagram below are the data from all nitrates except 800 and 1600. Complete the scatter diagram by plotting the points corresponding to these two nitrates.

2. Examine the above scatter diagram and describe the nature of the relationship between Absorbance and Nitrate.

3. Which statistical technique can be used to model the relationship between income and test score?

4. Figure 1 below is a screen dump of the output from a simple linear regression (SLR) analysis of the data.

Examine its contents carefully, and then complete the SLR dialogue box in the screen dump in Figure 2 on the next page.

[pic]

Figure 1. Screen dump of output from simple linear regression on data.

Figure 2. Screen dump of worksheet containing data and PHStat SLR dialogue box.

5. What is R2 for this data? Write a sentence to explain what it means.

6. Calculate the correlation between nitrates and absorbance. Note (use the information in Figure 1.)

r = ________________

Is this what you expected? Explain.

7. Using the information contained in Figure 1, write down the least-squares regression line for this data.

8. Draw the least-squares regression line on the scatter diagram.

9. Complete the following table by calculating the predicted absorbances and corresponding residuals for nitrates 800 and 1600. (note: the values given where calculated on Excel –ie with no intermediate rounding)

|Nitrate |Absorbance | |Residual |

|(mg/litre water) | |Predicted Absorbance | |

| | |absorbance | |

|50 |7.0 |7.32 |-0.32 |

|50 |7.5 |7.32 |0.178 |

|100 |12.8 |12.99 |-0.19 |

|200 |24.0 |24.31 |-0.32 |

|400 |47.0 |46.98 |0.02 |

|800 |93.0 | | |

|1200 |138.0 |137.62 |0.38 |

|1600 |183.0 | | |

|2000 |230.0 |228.26 |1.74 |

|2000 |226.0 |228.26 |2.26 |

10. Plot the residuals for nitrates 800 and 2000 on the residual plot below.

11. [pic]

Considering all the above information discuss whether you consider this to be an appropriate model to use.

12. In the context of this particular problem, interpret:

a) the slope

b) the intercept

13. Discuss the dangers of predicting beyond the range of the data.

Some of my favourite sources of fresh inspiration:

Books

Lovatt, C., Lowe, C., (1993) Chance and Data Investigations Volumes 1 and 2, Curriculum Corporation

Not new and often geared to younger students but I find they are a great starting point for interesting ways to introduce and explore new (or ‘old’!) topics.

Garfield J (ed) (2005) Innovations in Teaching Statistics , The Mathematical Association of America.

“This is a book of stories about teaching statistics” I’ve only just found this bk – it appears to be mainly about teaching ‘Introductory Statistics courses in University – tho that means some is applicable to Year13.

Gelman,A. Nolan, D (2002), Teaching Statistics – a bag of tricks Oxfod University Press.

Loads of ideas – includes both demonstrations and project type stuff- geared at developing ‘thinking’!!

Green,D. (Ed) (1994) Teaching Statistics at its Best, Teaching Statistics Trust

A compilation of of the best articles in Teaching Statistics (Journal) from volumes 6-14. – some interesting stuff – but not much that’s both ‘new and useful’ – the good ideas have been well shared around in the intervening years

Hawkins, A., Joliffe, F., Glickman, L., (1992) Teaching Statistical Concepts. Can be a bit heavy in parts – but has some good ideas

Journals

Journal of Statistics Education

More geared to university teaching – but ‘free’ on line

Mathematics in Schools - published by the Mathematical Association (UK)

Some articles are available on line at:



Teaching Statistics - to view recent copies need to subscribe – but some articles can be viewed – along with contents of current journals at



mostly geared to teaching in schools

Websites

**Use

My favourite for data with a story is DASL

My favourite - Simulations:

I especially like the Central limit theorem one (there is a worksheet to go with this on **

Data set included in excel file – from DASL (see web address above)

Datafile Name: Chromatography

Datafile Subjects: Science

Story Names: Chromatography

Reference: Moore, David S., and George P. McCabe (1989). Introduction to the Practice of Statistics. Original source: Kurtz, D.A. (ed), Trace Residue Analysis, American Chemical Society Symposium Series No. 284, 1985, Appendix.

Authorization:

Description: Results of a study of gas chromatography, a technique which is used to detect very small amounts of a substance. Five measurements were taken for each of four specimens containing different amounts of the substance. The amount of the substance in each specimen was determined before the experiment. The response variable is the output reading from the gas chromatograph. The purpose of the study is to calibrate the chromatograph by relating the actual amount of the substance to the chromatograph reading.

Number of cases: 20

Variable Names:

1. amount: amount of substance in the specimen (nanograms)

2. response: output reading from the gas chromatograph

Abstract: Results of a study of gas chromatography, a technique which is used to detect very small amounts of a substance. Five measurements were taken for each of four specimens containing different amounts of the substance. The amount of the substance in each specimen was determined before the experiment. The response variable is the output reading from the gas chromatograph. The purpose of the study is to calibrate the chromatograph by relating the actual amount of the substance to the chromatograph reading.

Figure 1 shows a plot of response vs. amount with a regression line superimposed. Note the large range of x-values. The regression of amount on response has an R-square of 99.9%. However, despite this, the plot shows that the regression line passes through the data only for the largest amount tested. The responses for other amounts either lie completely above or below the line.

Figure 2 shows this more clearly, and also shows that the variability of the residuals increases as the predicted values increase. These are failures of the regression assumptions of linearity and constant variance. They make the results of the regression suspect. Despite appearances, the data do not seem linear. Perhaps transforming the data would provide a better fit.

-----------------------

A manufacturers reputation and sales are very dependent on delivering what’s advertised – lots of chocolate chips – BUT economics suggest that costs (chocolate chips ) need to be kept to a minimum

How many chocolate chips do we need to ensure a reasonable number per biscuit

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download