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• Probability

Probability is used to predict the likeliness of something happening. It is always given between 0 and 1. An experiment is a repeatable process that can have a number of outcomes An event is one single or multiple outcomes A sample space is the set of all possible outcomes For two events, E₁ and E₂, with probabilities P₁ and P₂ respectively: To find the probability of either E₁ or E₂ happening, add the two probabilities, P = P₁ + P₂ To find the probability of both E₁ and E₂ happening, multiply the two probabilities, P = P₁ x P₂ The sample space for rolling two fair six-sided dice and adding up the numbers that show would look like this: To work out the probability of getting a particular result, you count how many times the result occurs and divide by the total number, 36 (since 6² = 36). So to work out the probability of getting a 10, count the number of tens and divide by 36: 3/36 = 0.0833 Generally, give your answers as decimals to three significant figures Conditional Probability If the probability of an event is dependent on the outcome of the previous event, it is called conditional. Conditional probability is noted using a vertical line between the events: The probability of B occurring, given that A has already occurred is given by P(B|A) For two independent events: P(A|B) = P(A|B') = P(A) Experiments with conditional probability can be calculated using a two-way table/restricted sample space: Venn Diagrams A Venn diagram is used to represent events happening. The rectangle represents the sample space, and the subsets within it represent certain events. Set notation is used to describe events within a sample space: A ∩ B represents the intersection A ∪ B represents the union Adding a dash, ', means the compliment, or "not" The addition formula is very useful: P(A ∪ B) = P(A) + P(B) - P(A ∩ B) Mutually Exclusive Events As you can see from the Venn diagrams, mutually exclusive events do not intersect. This means there is no overlap, so: P(A or B) = P(A) + P(B) Independent Events When one event has no effect on the other, the two events are described as independent. Therefore: P(A ∪ B) = P(A) x P(B) Conditional Probability in Venn Diagrams You can find conditional probability easily from Venn diagrams using the multiplication formula: P(B|A) = P(B ∩ A) / P(A) Tree Diagrams Tree diagrams are used to show the outcomes of two or more events happening, one after the other. For example, if there are 3 red tokens and 7 blue tokens in a bag, and two are chosen one after the other without replacement (the first is not put back into the bag), a tree diagram can model this: When you have worked out the probability of each branch, add them together - if they sum to 1, it is correct. Conditional Probability in tree diagrams Tree diagrams show conditional probability in their second and third etc columns. The multiplication formula still applies: P(B|A) = P(B ∩ A) / P(A)

• Correlation & Regression

Bivariate data is data with two variables, and can be represented in a scatter diagram. We can describe the correlation between the two variables based on how much of a straight line the points on the diagram form. Correlation describes the nature of the linear relationship between two variables. A negative correlation occurs when one variable increases as the other decreases. A positive correlation occurs when both variables increase together. Causation The relationship can be described as causal if a change in one variable induces a change in the other. It is vital to remember that just because there may be a correlation, no matter how strong, between two variables, it does not mean the relationship is causal. Correlation does not imply causation You need to consider the context of the variables and use common sense to decide whether or not there is causation as well as correlation. Measuring Correlation The product moment coefficient, r, is a measure of strength for linear correlation between two variables. It takes values from -1 to 1, where If r = 1 the correlation is perfect and positive If r = 0 there is no correlation at all If r = -1 the correlation is perfect and negative You calculate the product moment coefficient using a stats-equipped scientific calculator. On a CASIO ClassWiz fx-991EX, to calculate the product moment coefficient, r: Click MENU Click 6: statistics Click 2: y=a+bx Input your data in the table Click AC Click OPTN Click 3: Regression Calc r is the product moment coefficient Linear Regression The line of best fit on a scatter diagram approximates the relationship between the variables. The most accurate form of line of best fit is the least squares regression line, which minimises the sum of the squares of the distances from each data point to the line. The regression line is plotted in the form y = a + bx Where b tells you the change in y for each unit change in x. If the correlation is positive, so is b, and vice versa. To calculate a and b, use your calculator and follow the steps above for the product moment coefficient. Independent & Dependent Variables The independent variable is the one that is being changed, the dependent variable is the one being measured and recorded. The independent variable should always be plotted on the x-axis The dependent variable should always be plotted on the y-axis You should only ever use the regression line to make predictions for the dependent variable Exponential Models Exponentials and logarithms can be used to model non-linear data that still has a clear pattern. If the equation is in the form y = axⁿ, a graph of log(y) against log(x) will give a straight line where log(a) is the y intercept and n the gradient. If the equation is in the form y = ab^x, a graph of log(y) against x will give a straight line where log a is the y intercept and log b the gradient.

• Statistical Distributions

• Hypothesis Testing

A hypothesis is a statement that has yet to be proved. In statistics, the hypothesis is about the value of a population parameter, and can be tested by carrying out an experiment or taking a sample of the population. The test statistic is the result of the experiment / the statistic generated from the sample. In order to perform a hypothesis test, two hypotheses are required: The null hypothesis, H₀ is the one you assume to be correct The alternative hypothesis, H₁ is the one you are testing for, to see if the assumed parameter is correct or not. A specific threshold for the probability of the test statistic must also be defined. If the probability of the test statistic is lower than this threshold, there is sufficient evidence to reject H₀. If it is above the threshold, there is insufficient evidence to reject H₀. This threshold is known as the significance level, and is typically set at 1, 5 or 10%. When ending a hypothesis test, you must conclude by saying whether or not there is sufficient evidence to reject H₀. Do not say accept or reject H₁ Critical Regions & Values If the test statistic falls within the critical region, there is sufficient evidence to reject H₀. The critical value is the first value to fall inside the critical region. The acceptance region is the set of values that are not in the critical region, so there is insufficient evidence to reject H₀. The actual significance level is the probability of incorrectly rejecting the null hypothesis. What this actually means is that: the actual significance level is the probability of getting the critical value One- and Two-Tailed Tests Hypothesis tests can be one-tailed or two-tailed. This refers to how many critical regions there are: For a one-tailed test, H₁: p < ... or H₁: p > ... and there is only one critical region For a two-tailed test, H₁: p ≠ ... so there are two critical regions, one on each 'tail' See the examples below. Hypothesis Tests on Binomial Distributions Often, hypothesis tests are carried out on discrete random variables that are modelled with a binomial distribution. One-Tailed Example A discrete random variable, X, is distributed as B(12, p). Officially, X is distributed with a probability of 0.45. However, there is a suspicion that the probability is, in fact, higher. Find, at the 5% significance level, the critical region and actual significance level of the hypothesis test that should be carried out. Write out the hypotheses & test statistic H₀: p = 0.45 H₁: p > 0.45 X∼B(12, p) Since we are only looking at whether or not the probability is more than 0.45, it is a one-tailed test. Therefore, look for the first value of X for which the cumulative probability is more than 0.95 (1 - 0.05, the 5% significance level) As you can see, the first value to have a cumulative probability of more than 0.95 is 8, so: The critical value is 8 The critical region is > 7 Find the actual significance level 1 - 0.964 = 0.036 0.036 = 3.6 % The actual significance level is 3.6% Write a conclusion If the experiment were repeated 12 times, and 8 or more of the 12 trials were successful, there would be sufficient evidence to reject H₀, suggesting the probability is indeed higher than 0.45 Two-Tailed Example a. A manufacturer of kebab-makers (a kebab-maker-maker, if you will) claims that just 25% of the kebab-makers he makes make low quality kebabs. At the 10% significance level, find the critical region for a test of whether or not the kebab-maker-maker's claim is true for a sample of 12 kebab-makers. Write out the hypotheses and test statistic H₀: p = 0.25 H₁: p ≠ 0.25 X∼B(12, p) We do not know if the probability could be more or less than 0.25, so the test is two tailed. Therefore, divide the significance level by two, and find the critical region. This will be any cumulative probability that is less than 0.05 or more than 0.95 Here you can see the critical region is in two parts, one at each 'tail' of the values; The critical region is X < 1, X > 5 b. A random sample of 12 kebab-makers is taken, and 5 are found to make low quality kebabs. Does this imply the kebab-maker-maker is lying? Method 1 See if 5 is in the critical region 5 is not > 5 not < 1 Conclude 5 does not lie within the critical region for this test (X < 1, X > 5), so there is insufficient evidence to reject H₀ - this implies the kebab-maker-maker is not lying. Method 2 Find the cumulative binomial probability when X=5 When X∼B(12, 0.25), P(X=5) = 0.946 Conclude P(X=5) = 0.945, which is not within the significance level for the test. Therefore, there is insufficient evidence to reject H₀ - this implies the kebab-maker-maker is not lying. Hypothesis Tests on Normal Distributions You can carry out hypothesis tests on the mean of a normally distributed random variable by looking at the mean of a random sample taken from the overall population. To find the critical region or critical value, you need to standardise the test statistic: Then, you can use the percentage points table to determine critical regions and values, or you can use the inverse normal distribution function on a scientific calculator. Example The kebabs that the kebab-maker makes have diameter D, where D is normally distributed with a mean of 4.80 cm. The kebab-maker is cleaned, and afterwards a 50 kebabs are made and measured, to see if D has changed as a result of the cleaning. D is still normally distributed with standard deviation 0.250 cm. Find, at the 5% significance level, the critical region for the test. Write out your hypotheses H₀: μ = 4.8 H₁: μ ≠ 4.8 Assume H₀ is true: Sample mean of D, Ď ∼ N(4.8, 0.25²/50 ) Code data: Z = (Ď - 0.48) / (0.25/√50) Z ∼ N(0, 1) The test is two tailed, so area on each side should be 0.025 (half of 5%): Decode, using ±1.96 (Ď - 0.48) / (0.25/√50) = -1.96 Ď - 0.48 = -0.0693 Ď = 0.411 (Ď - 0.48) / (0.25/√50) = 1.96 Ď - 0.48 = 0.0693 Ď = 0.549 Conclude The critical region is when the sample mean is smaller than 0.411 or larger than 0.549 Hypothesis Tests for Zero Correlation You can determine whether or not the product moment coefficient, p, of a sample indicates whether or not there is likely to be a linear relationship for the wider population using a hypothesis test. Use a one-tailed test if you want to test if the population p is either > 0 or < 0 Use a two tailed test if you want to see that there is any sort of relationship, so p ≠ 0 The critical region can be determined using a product moment coefficient table. It depends on significance level and sample size. To calculate the product moment coefficient of the sample, use your calculator (see notes sheet on regression & correlation).

• Indices & Algebraic Methods

• Sequences & Series

A sequence is a list of numbers with a particular relation; a series is the sum of such a list of numbers. Sequences are sometimes a,so referred to as progressions. Arithmetic Arithmetic Sequences An arithmetic sequence has a constant defined distance between terms, e.g. 1, 3, 5 7, 9 etc. The first term is 1 and the common difference is +2. The common difference can be positive (the sequence is increasing) or negative (the series is decreasing). To calculate the nth term, u(n), of an arithmetic sequence: where a is the first term and d the common difference. Arithmetic Series An arithmetic series is the sum of all numbers in an arithmetic sequence. The sum of the first n terms is given by: where a is the first term, d the common difference, and l the last term. Geometric Geometric Sequences A geometric sequence is a sequence where there is a common ratio, not a common difference. This means the relationship between the numbers is a multiplication, not addition/subtraction. For example, the sequence 2, 4, 8, 16, 32 etc. is geometric - each term is multiplied by 2. The formula for the nth term of a geometric sequence is: where a is the first term and r is the common ratio. Geometric Series A geometric series is the sum of the first n terms of a geometric sequence. The sum of the first n terms is given by: where a is the first term and r the common ratio. The common ration cannot be 1, else each number would be the same and the sum is just n x a. Sum to Infinity As n tends towards infinity, the sum of the series is called the sum to infinity. If a series is getting bigger, its sum tends to infinity, e.g. the series 2 + 4 + 8 + 16 + 32 etc. This happens when r > 1, and the series is known as divergent. If a series is getting smaller, its sum tends to a finite value, e.g. the series 2 + 1 + 1/2 + 1/4 + 1/8 etc. This happens when r < 1, and the series is known as convergent. When a series is convergent, we can calculate the fixed value that its sum tends to, its sum to infinity: Sigma Notation The Greek capital letter sigma, ∑, is used to note sums. Limits are shown above and below the ∑ to tell you from which term to which term to sum, followed by an expression. This is the function used to calculate every term in the sequence: There are standard results for this you can substitute into other series: If the series is given in the form of an expression, but there are too many terms to write out, just write out the first few to find the first term and common difference/ratio. Then use standard arithmetic/geometric sum equations. Recurrence Relations A recurrence relation defines each term of a sequence as a function of the previous term. This means you need to know at least one term in a sequence to work forwards or backwards from. A recurrence relation is noted as: An example of how to calculate a sequence from a recurrence relation is: There are three forms a sequence can take, depending on the recurrence relation: A sequence is increasing when u(n+1) > u(n) A sequence is decreasing when u(n+1) < u(n) A sequence is periodic if the terms repeat in a regular cycle: u(n+k) = u(n) where k is the order of the sequence. A sequence can also take none of these three forms. 1, 3, 5, 7, 9... is increasing 6, 4, 2, 0, -2... is decreasing 1, 3, 5, 1, 3, 5, 1, 3, 5... is periodic with an order of 3 1, 7, 4, -9, 12... is none of the above.

• Quadratic & Simultaneous Equations and Inequalities

• Graphs, Functions & Transformations