Definitions for S1

Statistical Experiment A text/investigation/process adopted for collecting data to provide evidence for or against a hypothesis. “Explain briefly why mathematical models can help to improve our understanding of real world problems” Simplifies a real world problem; enables us to gain a quicker / cheaper understanding of a real world problem Advantage and disadvantage of statistical model Advantage : cheaper and quicker Disadvantage : not fully accurate “Statistical models can be used to describe real world problems. Explain the process involved in the formulation of a statistical model.” • Observe real-world problem • Devise a statistical model and collect data • Compare and observe against expected outcomes and test model; • Refine model if necessary. A sample space A list of all possible outcomes of an experiment Event Sub-set of possible outcomes of an experiment.

Normal Distribution Bell shaped curve symmetrical about mean; mean = mode = median 95% of data lies within 2 standard deviations of mean 2 conditions for skewness Positive skew if ( Q3 − Q2 ) − ( Q2 − Q1 ) > 0 and if Mean − Median > 0 . Negative skew if ( Q3 − Q2 ) − ( Q2 − Q1 ) < 0 and if Mean − Median < 0 .

Independent Events P ( A ∩ B) = P( A) × P( B) Mutually Exclusive Events P( A ∩ B) = 0 Explanatory and response variables The response variable is the dependent variable. It depends on the explanatory variable (also called the independent variable). So in a graph of length of life versus number of cigarettes smoked per week, the dependent variable would be length of life. It depends (or may do) on the number of cigarettes smoked per week. Copyright www.pgmaths.co.uk - for GCSE, IGCSE, AS and A2 notes

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Data

Discrete Discrete data can only take certain values in any given range. Number of cars in a household is an example of discrete data. The values do not have to be whole numbers (e.g. shoe size is discrete). Continuous Continuous data can take any value in a given range. So a person’s height is continuous since it could be any value within set limits. Categorical Categorical data is data which is not numerical, such as choice of breakfast cereal etc.

Data may be displayed as grouped data or ungrouped data. We say that data is “grouped” when we present it in the following way: Weight (w) 6570Or Score (s) 5-9 10-14 Frequency 2 5 Frequency 3 7

NB: We can group discrete data or continuous data. We must know how to interpret these groups, So that Weight (w) 6570Or Score (s) 5-9 10-14 4.5 ≤ s < 9.5 9.5 ≤ s < 14.5

65 ≤ w < 70 70 ≤ w < 75

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⎛ 18 − 20 X − 20 23 − 20 ⎞ P (18 < X < 23) = P ⎜ < < ⎟ = P ( −1 < Z < 1.5 ) . 2 2 ⎠ ⎝ 2 If we now had a set of tables showing us all possible values for P ( Z < z ) then we could calculate this since P ( −1 < Z < 1.5 ) = P ( Z < 1.5 ) − P ( Z < −1) .

= So we have two curves (1) N (µ , σ 2 ) is the general normal distribution σ 2 (the variance ). We use the variable X. For example,

with parameters

µ (the mean)

and

(2) N (0 ,1) is the standard normal distribution with parameter 0 (the mean) and. 1 (the variance). We use the variable Z to distinguish it form the general normal case.

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Standard Normal Distribution The cumulative distribution function for the random variable Z is written as Φ(z) .

In other words Φ ( z ) = P( Z < z ) =

1 2π

−∞

∫e

z

− 1t2 2

dt .

From the tables we have Φ ( 0 ) = 0.5, Φ (1) = 0.8413, Φ ( 2 ) = 0.9772, Φ ( 3) = 0.9987 etc.

We see from the above that to calculate Φ(z) when z < 0 we use symmetry, i.e. we use...