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AS and A Level: Probability & Statistics

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Statistical diagrams

1. 1 When working with grouped data, if the class is from 9 to 12, this includes values from 8.5 up to 12.5, which means the class width is 4, not 3.
2. 2 In a histogram, the area is the frequency. The y-axis is the frequency density.
3. 3 When working out lengths of scaled histograms, it is always helpful to draw the rectangle and label the relevant sides with the lengths given.
4. 4 When drawing a stem and leaf diagram, make sure to include a key. The key is worth a mark. For example 2|1 represents 2.1 or on a different stem and leaf, 3|2 represents 32.
5. 5 Always draw a scale when drawing a box plot, the scale is worth a mark.

How to tackle questions on regression and correlation

1. 1 When asked about the relationship in a regression model, always get the context the correct way round. For example, weight does not affect height, height effects weight.
2. 2 When asked if your answer is reliable for the regression model, comment on whether the x value you used to get the answer is within the original data set. If the x value is within the boundary it is suitable. Never extrapolate when using a regression model.
3. 3 If you have found a regression model for a relationship between h and p, and are then told h=x+100 and p=y-20 and asked to find a regression model for x and y. Sub x+100 and y-20 into your original equation and re-arrange.
4. 4 When data is coded the correlation co-efficient is not changed.
5. 5 If a regression model is created using for example, heights and weights of children. This model could not be used to predict the weight of an adult. Models are very specific to the data with which they were created..

Normal distribution

1. 1 When answering normal distribution questions always draw a picture and shade in the part of the graph that you know and/or want.
2. 2 In a normal distribution, the area under the curve represents the probability.
3. 3 A normal distribution model is appropriate if the mean and median are the same, or very close.
4. 4 The big normal distribution table gives area to the left of the line. The small table has areas to the right of the line.
5. 5 If unsure what the question is asking. Do the first step which is to rewrite the question, but converted to the normal distribution.

1. Forensic Examination of Drugs by Thin Layer Chromatography.

or alumina (Al2O3) coated on an aluminium or plastic sheet. The plate constitutes the stationary phase. The sheet is then placed in a chamber containing a small amount of solvent, which is the mobile phase. The solvent gradually moves up the plate via capillary action, & it carries the deposited substances along with it at different rates due to the differential solubility of each of its components. The desired result is that each component of the deposited mixture is moved a different distance up the plate by the solvent.

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2. "Does wearing glasses have an effect on the pupils math's ability?"

Within these groups I must find the number of pupils who wear and don't wear glasses. Calculations: Total population boys: 131 girls: 108 Percentage boys: 131/239*100=55% girls: 108/239*100=45% Sample size: 80 boys:44 girls:36 My sample size is 80. I am using stratified sampling so I need the following number of pupils who wear and don't wear glasses. 22 boys who wear glasses 22 boys who don't wear glasses 18 girls who wear glasses 18 girls who don't wear glasses I needed to pick these to get a fair and unbiased representation of the overall population.

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3. Application of number: level 3 - Is House Buying a Good Idea or Not?

* Range: highest house price - lowest house price = �84,950 - �18,500 = �66,450 (to the nearest �) Other houses: * Mean: Sum of house prices / 30 = �3,334,600 / 30 = �111,153 (to the nearest �) * Range: highest house price - lowest house price = �249,950 - �32,500 = �217,450 (to the nearest �) All houses: * Mean: Sum of house prices / 60 = �4,919,045/ 60 = �81,984 (to the nearest �) * Range: highest house price - lowest house price = �249,950 - �18,500 = �231,450 (to the nearest �)

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4. Collect data with a view to estimating population parameters using estimation techniques.

The Central Limit Theorem Because I don't know anything about how the population is distributed I have to use the Central Limit Theorem. Even if you don't know how the parent population is distributed the central limit theorem allows you to make predictions as to the distribution of the sample means. Also with a large enough sample the sample mean will be close to the population mean. The central limit theorem says that: * If you take enough samples then the means will be normally distributed. * The mean of the sample means is approximately equal to the population mean.

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5. Standard addition was used to accurately quantify for quinine in an unknown urine sample containing approximately 100 &igrave;g cm&#8209;&sup3; of quinine.

In dilute solutions it has an astringent taste and is added to some types of tonic water. The analysis of quinine in urine is important in forensic science as quinine is frequently used as an adulterant in illicit heroin samples. Its presence can therefore be tested for in order to determine the presence of heroin in the body. Fluorescence and Phosphorescence Fluorescence and phosphorescence are phenomena associated with transitions between more than one excited state for a species. After excitation to a higher level, an electron drops by a non-radiative process to an intermediate level and then to the ground state giving rise to emission at a longer wavelength than that of the exciting radiation.

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6. Statistical Process Control.

Many of the techniques developed by mathematical statisticians for the analysis of data may be used in control of product quality. The expression "Statistical Process Control" may be used to cover all uses of statistical techniques for this purpose. Statistical Process Control is a method of monitoring a process during its operation in order to control the quality of product while they are being produced rather than relying on inspection to find problems after the fact. It involves gathering information about the product, or the process itself, on a near real-time basic so that the operator can take action on the process.

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7. Explain the proposition that it does not matter to a seller which type of auction is used.

First Price Sealed Bidding auctions differ from the auctions mentioned above because they are "closed" auctions. In this instance, each bidder submits a sealed bid (without knowing the bids of others) by a designated deadline. The bids are opened (at a specified time), and in the case of First Price Sealed Bidding, the item is awarded to the highest bidder at a price equal to that individual's bid. Second Price Sealed Bidding also sees the item going to the highest bidder, but at a price equal to the second highest bid. For each of these different types of auction, economists are concerned with finding the bidding strategies of rational bidders, and to determine the outcome of the auction, which involves determining who gets the item on offer and at what price.

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8. One basic assumption of Black-Scholes model is that the stock price is log-normally distributed with constant volatility. However, in option market, does this assumption hold?

In this way, we can capture the empirical distribution of stock price; its shape must be more accurate and accordingly more likely to be the same as the distribution of the company's share price in the real world. (Precisely, if we can employ more lognormal distributions to obtain the possibility for the movement of share price, we would get better distribution to describe real world.) Therefore, we could simply test the accuracy of Black-Scholes model by comparison. About Data: We chose Six Continents as our target company to do our analysis and made a comparison between its mixlognormal distribution and Black-Scholes lognormal distribution.

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9. Investigatings heights of girls and boys in year seven.

People may also have added their hair into their height. If I were collecting the data myself, I would have been able to ensure that the data was all collected absolutely accurately. Other things that could have biased my results are the fact that the data So I can begin to find averages and draw conclusions from my data, I need to take a sample of the records. There are three types of sample I could take to find the data records I want to use in my coursework; a Simple Random Sample, a Systematic Sample and a Stratified Sample.

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10. Mating choice and Lonely Hearts Columns.

But those who were middle aged tended to seek those who were of their age group. In present day it's more of being able to live with someone who can make you happy and give you and your sprouts a good chance in life. TERMINOLOGY Lonely-Hearts Column - relating to people who are looking for companions or marriage partners Financial Resources - money/wealth Personal Appearance - physical attributes Quota sampling - before the sample is selected the population is divided into appropriate categories from which the sample is selected.

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11. Factors influencing girls athletic performance throughout secondary school.

Hypothesis One - THE BETTER ONE IS AT SHOT PUT, THE BETTER THEY ARE AT DISCUS I hypothesise that the farther one can throw the shot, the farther one can throw the discus. To test my hypothesis, I am using data from 7H - 1998, 7K- 1999, 7A- 2000 and 7P - 2001 to test my hypothesis. I would eventually like to develop this hypothesis to compare the classes in the events of shot putt and discus, in an attempt to determine which year 7 class was the best at these activities.

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12. Investigate the prices of second hand cars and the factors that may affect the price.

First of all I calculated the % depreciation of all the cars in my sample. I calculated the % rather than straight monetary loss because it helps smooth out the extremes between car makes eg Bentleys and Ford. I did a scattergraph (appendix 2) of age against % depreciation and then drew a line of best fit. This showed a strong positive correlation that the older the car the more value it loses. This shows that my Hypothesis 1 is correct. COLLECTING DATA 2 After the first analysis I then looked at my sample again and found that the range was too big to draw any meaningful conclusions. I decided to change my sampling method.

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13. In this project I will be investigating three main hypotheses, which are all based on house prices.

A much more pleasant place to live, with mostly nice whether. I am very aware that the figures are the actual asking prices for the properties and so therefore the actual price of the property will either be more or less then the asking price. Naturally the actual sale prices for individual properties will vary depending on factors such as the size of the property, number of bedrooms, the size of the bedrooms (with an en-suite) number of bathrooms, whether it has a garden, garage, the quality of the house, the type of houses e.g.

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14. Will the population and community of plants increase upon progressing inland from the beach?

Also we are going to create a dune profile, to observe how the dunes naturally exist. How We Are Going To Do This... * We are going to use Ranging Poles and Clinometers (explained in more detail later) to record angles of depression and elevation. * We will use a tape to measure 200 meters inland from the beach and make observations every 10meters. * At every 10meter intervals we will take the pH of the soil using an analogue pH meter. * Also at every 10meter intervals we will place quadrates down to observe the percentage cover of certain species of plants (a greater detailed explanation later)

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15. What factors made a person better at estimating the size of an angle or the length of a line.

the percentage error of each group again using the calculation shown in the preliminary testing this was (difference between original and average estimates / actual size/length) * 100 But first we thought that in the data there may be rogue results these are called outliers and are values that do not follow the data in a reasonable trend and so can be eliminated using a certain formula that creates upper and lower fences and if values fall outside of these two fences they can be classed as outliers and will be dismissed from the data .

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16. Is there a correlation between happiness and sociability?

'glad, content, happy'(*4) There is lots of evidence that points towards happiness and sociability being related. These quotes show this relationship: "Social science surveys have universally concluded that people claim to be most happy with friends and family, or just in the company of others"(*5). This shows how people who socialise are likely to get a feeling of happiness; therefore, in theory the more sociable a person, the happier they should be. "Relationships make us extremely happy when they go well, and very depressed when they don't work out"(*5). This shows how the breakdown of social relationships can be the cause of unhappiness, but these relationships can also cause a person to be happy when they are going well.

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17. Which three factors affect the price of a second hand car.

Age - This tells you how long the car has been running, if it's been running a long time its parts may be worn from the rust and may need a repair. 3. Engine Size - Many people want a large engine, this may be because they want to go fast or maybe they live up a hill and want to make sure that it can go up without much trouble. 4. Style - The style is very important because it affects whether you want to buy the car or if it suits you.

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18. Statistics - My aim is to investigate whether it is possible to gain information about authorship of a text by using statistical measures.

However, the disadvantage is the results may include natural variation or bias and so may not be representative of the whole population and it may not be accurate. There are rules that must be followed when choosing a sample. > The sample size must be large enough so that the results are more accurate. A very small sample may not represent the rest of the population. So I must make sure that any sample I take is large enough to be representative of the population as a whole.

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19. In this investigation my aim is to find the mean height of boys and girls in year 10.

It is computed as the average squared deviation of each number from its mean. For example, for the numbers 1, 2, and 3, the mean is 2 and the variance is: ?2 =. The formula for the variance in a population is Where ? is the mean and N is the number of scores. Standard Deviation The formula for the standard deviation is very simple: it is the square root of the variance. It is the most commonly used measure of spread. An important attribute of the standard deviation as a measure of spread is that if the mean and standard deviation of a normal distribution are known, it is possible to compute the percentile rank associated with any given score.

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20. The average pupil.

These will include scatter graphs, histograms, cumulative frequency graphs, bar charts and box plots. These will allow me to investigate the spread of the data in an ordered manner. When I have to solve the hypotheses, I will outline the relevant hypothesis and then I will make a list of the pupil's names and their scores in the areas connected to the hypothesis (only for the last two). I will then examine the data, and compare it with the use of charts. Finally I will come to a conclusion by combining all the gathered data from all my hypotheses and I will answer the question, what is the average child.

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21. Investigate the distribution of males&acute; and/ or females in families. You may choose, for example, to collect data on the distribution of girls in families of three children and to estimate the probability of a female birth.

136 was placed into a graphic calculator which then randomly gave out 30 numbers. It should be noted that the larger the sample size the more reliable the results are likely to be. Hypothesis: The probability of having a boy in a three-child family is above 75 %. The probability of giving birth to a boy is the same as a girl, this can be proved by using genetic code which is below. x y Boy xy x xx xy When these two are mixed there are 4 out comes: Girl xx x xy xx As the probability of a boy and girl has been proved equal a suitable probability model can be chosen.

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22. We were set a piece of coursework that involved asking people to guess the line and the angle size that were drawn on a piece of paper.

Hypothesis 1: I think that more people will do better at guessing the angle because you can work out 90 degrees and then guess the last bit. Estimating a line is much harder as it is harder to imagine a centimetre or say 15 centimetres. Hypothesis 2: I think that girls will be better at estimating than boys estimate as girls are normally more careful then boys. They will probably take more time to estimate, so there estimate will be more accurate.

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23. House buying - a good idea or not?

Three house prices were be obtained from each of the 10 estate agents - one of which would be the highest house price whilst another would be the lowest house price, and one of the house prices was selected from the mid-range. This method of collecting information was an attempt to ensure that both samples were reliable due to consisting of a wide variety of data. A table to show the data collated for the samples for other houses and first-time buyers' houses (in ascending order): Other houses Price (�)

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24. Statistics - Men are more accurate than women at estimating the length of a line are.

This would make my project too long and it doesn't link with my first two hypotheses. I am going to collect my results using a tally chart as this will show my results in a simple way and I will be able to compare the results to see whether my hypothesis is correct. I am going to do this for both the line and the angle results. Tally Chart for men estimating the length of the line. Length of Line (L)

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25. The Average Student Plan.

Another reason for selecting this data collection is that the data is easily accessible. There are two hundred and seven pupils in my year and obtaining their measurements will not be a problem. I will hopefully have the opportunity to compare my results with the National Data from the Child Growth Foundation (1996), which my teacher has recommended we find and use because using this I will be able to compare upper and lower quartiles and the medians. Firstly my aim is to research and discover some hypotheses about an average student using statistical techniques.

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