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# AS and A Level: Probability & Statistics

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Meet our team of inspirational teachers Get help from 80+ teachers and hundreds of thousands of student written documents ## 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.

• Marked by Teachers essays 2
1. ## Aim: in this task, you will investigate the different functions that best model the population of China from 1950 to 1995.

Linear trend line: * The above graph now has a linear trend line (with the equation of f(x) = 15.5x - 29690.25). Although it is very close to each point, it isn't perfect (for example, the data for 1950 and 1965 hardly touch the line). Furthermore, this model predicts that China's population would simply increase at a constant rate - which is unlikely since it is expected that a population would increase exponentially, in an ideal world. Exponential trend line: * The exponential trend line (equation: f(x) = 1.02^x) just about fits the data, but there are more anomalies than the linear line.

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2. ## Testing Materials Coursework

For this investigation we are going to use a clamp stand, a hook, a cotton reel, a clamp and weights. We are going to put the clamp stand on the bench and place the cotton reel, with the sample tied around it, in the clamp. We will then increase the load on the material until it breaks and finally we will measure the load needed. We will make sure that our work is safe by wearing safety goggles throughout the investigation and keeping the weights away from the bench so that

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3. ## Maths Cars coursework Plan

I will also need to see the correlations between Age and Price, and the correlations between, Mileage and Price. This will be done to see if they have a negative correlation or positive correlation. This will be done using scatter graphs and how strong each factor influences prices by spearman coefficient of rank correlation. My hypotheses for this investigation are: 1. There is a negative correlation between age and price. 2. There is a negative correlation between mileage and price.

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4. ## science resistance gcse

If we looked at other students in other years or in different areas that could of helped to support the hypotheses more strongly. There is one way that I could have been more accurate with my results and that would be to have a bigger sample size. This would make my results more accurate as it would be taking results of a lot more people which means its more accurate in saying what it would be like for a large sum of people but if you have a small sample size the results may not reflect the large sum of people which makes the sample inaccurate.

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5. ## Statisctics coursework

2000-2999 Ill 3 x 2500 = 7500 3000-3999 ll 2 x 3500 = 7000 4000-4999 III 3 x 4500 = 13500 5000-5999 Il 2 x 5500 = 11000 6000-6999 Il 2 x 6500 = 13000 7000-7999 Il 2 x 7500 = 15000 8000-8999 ll 2 x 8500 = 17000 9000-9999 0 x 9500 = 0 10000-10999 l 1 x 10500 = 10500 11000-11999 I 1 x 11500 = 11500 12000-12999 I 1 x 12500 = 12500 53 154500 Africa Mean =154500/53=2915.09 Africa Modal Group = 1000-1999 I am now going to find the cumulative frequency and use that data to

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6. ## A sociology study of male self image

His primary method was 40 structured questionnaires therefore the method used collected quantitative data and adopted a positivist approach. The research may have decided to adopt a positivist method to conduct his study because he wanted to compare the views of his peers those of a mixed group drawn from the adult's population of Chester. The main focus of his investigation was to evaluate and compare two distinct age groups. Access to the younger sample was easier to obtain than the older sample and where to draw the older sample from was more of a problem.

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7. ## Original Writing: Dramatic Monologue - Monologue of a Disturbed Personality

I wish for anyone else to please call me, Lizzie." I paused for a minute the silence was unbearable. I just wished he would break this terrible silence. It feels like the clocks have all stopped and somebody has flicked a switch and stopped the time in this room only. I can hear people outside talking and the cars on the nearby roads. I can hear the roaring of their engines. I can hear the ticking of the old grandfather clock in the corridor.

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8. ## MAYFIELD HIGH COURSEWORK

She needs this fat because her body is getting ready top bear children. Boys on the other hand don't put that much weight on in the early years because it is not needed but as they grow older they put on more weight on as muscle because boys are more powerful and stronger than girls because and their strength is needed. TYPES OF SAMPLING: Random Sampling: Random sampling is used to reduce bias so that a set of data is not one sided. Random sampling can be obtained by using a calculator. When using random sampling you must ensure that all samples are equally likely to be chosen. Stratified sampling: Strata means 'layer'.

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9. ## I intend to investigate the relationship between height and weight

The information that will be essential to collect for my investigation is height, weight, gender and year group. I will use a sample size of 7% of the population size, 1183 students, which calculates to be 83, rounded up to the nearest whole number. I had the option of selecting any sample size between 5% and 10%. Furthermore, I know that the bigger the sample the more useful the data will be, but I chose a small sample size because I believe that it will be easier to manage and to analyse.

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10. ## India PopulationPyramid for 1995

for the year 2000: India Population Pyramid for 2003 Age and s*x distribution for

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11. ## Sampling Techniques.

Suppose that the N units in the population are numbered 1 to N in some order. To select a systematic sample of n units, if then every k-th unit is selected commencing with a randomly chosen number between 1 and k. Hence the selection of the first unit determines the whole sample, e.g., N = 5,000, n = 250 therefore k = 5000/250 = 20. Therefore, select every 20th item commencing with (say) 6.

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12. ## Statistics S2 - Terms to learn

Mean The mean of random variable x is equal to the expected value E(x). This is the population mean and denoted by �. A sample mean x may be used to estimate �. Varience A measure of the dispersion of a random variable or of a sample. Is equal to E(x2) Statistic A function of a sample, a quantity calculated from a set of observations. E.g. sample mean and sample varience Poisson Distribution The discrete probability distribution with probability function given by P(X=r)

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13. ## Different factors that Influence Sociologist's choice of research methods.

Quantitative methods aren't so easy to quantify. Topics such as Bullying would involve qualitative research, because it is a detailed topic, which allows a deeper, and a valid relationship. Qualitative methods involve validity because the research gives depth, but cannot involve large numbers, therefore it can be unreliable. The researcher needs to decide upon the people to be studied, this is the sample. It is usually chosen in order to be representative of the particular population to be studied e.g.

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14. ## Investigate two categorical variables whose relationship seem interesting. The two categorical variables, which we have chosen to investigate is gender and smoking.

It should be noted that the larger the sample size the more reliable the results are likely to be. There are several types of chi-square test depending on the way the data was collected and the hypothesis being tested. We will begin with the simplest case: a 2X2 contingency table. When a comparison is made between one sample and another, a simple rule is that the degrees of freedom equal (number of columns minus one) X (number of rows minus one). For our data this gives us (2-1) X (2-1) equals 1. The table had percentages across. Comparison down the columns, suggest that men (33.3%) are less likely to smoke compared to women (47.4%)

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15. ## "Examine the role of the physical environment in the development and spread of malaria".

The key contributing factors that lead to the spread of malaria include moisture, warmth, and vegetation cover in order for the mosquitoes to flourish leading to the spread of the disease. Malaria is a deliberating infectious disease characterised by chills, shaking and intense fever. These symptoms often lead to death. Analysis Temperature Malaria needs temperatures of 16�C and above; otherwise the parasite is unable to develop inside the stomach of the mosquito. Above 32�C, the parasite is unlikely to survive.

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16. ## 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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17. ## 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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18. ## In this coursework I am going to investigate what influences the prices of a second hand car.

Secondly I will use standard deviation to mean all the second hand price of the entire car make I have selected. Ford This graph is a negative correlation and shows that as the age the ford cars increases the second hand price goes down rapidly. This graph is positive correlation and shows that as the engine size increases, the second hand price goes up by a small amount. Vauxhall This graph is a negative correlation and shows that as the age of the Vauxhall car increases the second hand price goes down This graph is a positive correlation and shows

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19. ## To investigate any relationship between height and foot size in adolescents, aged 7 to 16 years.

After considering this, I have decided to investigate foot size and height. Aim To investigate any relationship between height and foot size in adolescents, aged 7 to 16 years. Hypothesis The taller a person is, the longer their feet are. Plan Before starting my investigation, there are a few points I need to simplify. For example, is my measurements were different, would this be suitable to investigate? I need to ensure that both variables are measured in centimetres, so I can fairly compare them. If I am going to talk about taller or shorter, I need to state what is "tall" and what is "short".

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20. ## I chose to investigate the relationship between the height and weight of year 7 students. I think that as the height increases, so will the weight.

Using a computer, I opened the data using Microsoft Excel. As I was only investigating height and weight of year 7 students, I deleted all the irrelevant columns, such as year 8, 9, 10, 11 students, IQ, hair colour, eye colour etc. I then sorted the data, again using the computer. I discarded my outliers in the data. I discarded data which either had very tall or very heavy people on and this may be false data, unreliable and affect my final result. I needed a sample of 50 students and I had 200 year 7 students.

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21. ## I am going to estimate the population of woodlice in parts of the school ground, using Lincoln Index.

The woodlice are then released into their habitat and allowed to mix with the rest of the population. After two days, woodlice are then caught from the same location and the amount caught is recorded, along with the number of the woodlice that have already been marked from the first sample. Result Number of Woodlice Original sample, S1 35 Second sample, S2 39 Number marked in second sample, m 2 Lincoln Index

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22. ## To find out if the mileage that a car has run is related in any way to the percentage decrease in the price of the car from original price to secondhand price.

Method I will pick out the data I need for this project and take samples. Here I will need the statistics of mileage and both prices then use the spread sheet to work out the price decrease percentage. I will test them for any relation they may have and draw scatter diagrams. The scatter diagrams would have lines of best fit. A cumulative frequency table may also be useful in finding the relations. * Data The information I need to collect are the prices and the mileage. These are available in the excel spreadsheet from the KGV math site and I will be working form this.

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23. ## My Hypothesis is that 'Boys in Year 10 are better at estimating lengths of straight lines than Boys in Year 7'

STRATIFIED SAMPLING In a stratified sample, the population is first divided into categories and the number of members in each category is determined. The sample is then made up of these categories in the same proportions as they are in the population. Random sampling chooses the required numbers in each category. In a random sample, every member of the population has an equal chance of being chosen. To obtain a sample of 60 students stratified according to Year 10 and Year 7; I will have to calculate the number of boys from Year 10 and Year 7, which must be represented in the sample.

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24. ## Gcse Statistics: Coursework

I think this because, - For instance, if I compared a man of 45 and a boy of 15 to estimate, I'd predict that the man would estimate closest. I think this because the man has possibly got more experience than the boy at estimating. - A person's ability to estimate can be varied by intelligence. For example, if I asked two boys in Year 10 to estimate, I would expect the boy with the better mathematical knowledge to estimate closest.

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25. ## Research on Correlation

as the price of new cars falls, demand for new cars will tend to rise). Why do Businesses use it? Businesses may use correlation for number of reasons. They could use their statistics to anticipate how the market will react to a sudden change in supply and demand. For example, if the price of the average computer falls, there may be a slight or sudden change in consumer spending, however, depending on the market, the change may rise for those who wouldn't mind a cheaper computer, or on the other hand it could be classes as a luxury good, therefore the price may be important giving the goods an exclusive feel.

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