# Further statistics

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Bivariate data is data for two variables. Bivariate data is used to compare two sets of data and for finding a relationship between those two variables.

A regression line is the line that best fits the data. If the scatterplot of bivariate data shows a linear pattern and the correlation between the variables is very strong, a regression line can be drawn through the scatter points. The method of least squares can be used to find the y-intercept of a regression line. Using this method makes the total of the square of the errors as small as possible, and thus returns the best regression line. The gradient and the y-intercept can be found using the following formulas by the least squares method: m = (NΞ£(xy) β Ξ£xΞ£y)/(NΞ£(xΒ²) β (Ξ£x)Β²), where N is the total number of points, and c = (Ξ£y β mΞ£x)/N.

The discrete probability distribution of a random variable is the set of all possible outcomes it can take and its corresponding probabilities. For a discrete probability distribution, the probabilities must sum to one. One discrete probability distribution is the binomial distribution. The binomial distribution gives the probability of getting a number of successes in an experiment which is repeated for a number of trials. The probability for a binomial distribution is found using the following formula: P(X = x) = (nCx)pΛ£(1-p)^(n β x), where p is the probability of success, n is the number of trials and x is the total number of successes. The binomial distribution when there are only 2 possible outcomes, those being a success or a failure.

A normal distribution is a symmetrical bell curve. It is described by two parameters: the mean, which is the central maximum of the curve, and the standard deviation. The probability of a random variable that follows a normal distribution is found by calculating the area under the curve in the given range. For a given continuous random variable that follows a normal distribution, approximately 68% of all values lie within 1 standard deviation of the mean, 95% of all values lie within 2 standard deviations of the mean and approximately all of them lie within 3 standard deviations of the mean. When the parameters of the normal distribution are known, the calculator function for the normal distribution can be used to calculate probabilities.

A hypothesis test is a process that examines a population parameter value proposed by the null hypothesis compared to the alternative hypothesis at a given significance level. The null hypothesis, Hβ, is what is asssumed to be true unless proven otherwise. If there is significant enough evidence from the test statistic that is calculated using the given data, Hβ may be rejected in favour of the alternative hypothesis, Hβ. The critical region is the set of values for the test statistic for which the null hypothesis would be rejected in favour of the alternative hypothesis. A hypothesis test may either be one-tailed, which looks for either an increase or decrease of the parameter, or two-tailed, which looks for any change in the parameter. For a hypothesis test to be carried out, the probability of the observed test statistic coming from the given distribution is found. If this is smaller than the significance level, the null hypothesis is rejected.

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