| Question |
Answer |
Flap 3 |
| A population | is the set of all the possible items to be observed.
example: Whilst investigating the height of males in Wales, the population would be the height of all the males in Wales. | |
| Random sampling: | this method gives every item of the population an equal chance of selection. This can be done in various ways for example by simply picking out of a hat or by using a random number generator on a calculator. | |
| Stratified sampling: | some populations are naturally split into a number of strata (kind of like sub groups). We can separate the strata and find what proportion of the population is in each stratum. We can then select a random sample from each stratum proportional to its size | |
| measure of central tendency | is just a mathematical and rather posh way of saying "averages". | |
| The Mode | It is the piece or pieces of data that occur most often. | |
| The Median | The median is the middle piece of data when the data is in numerical order.->With 50 pieces of data, even, we must find halfway and the next value. In this case, the 25th and 26th values. The median will be halfway between these values. | |
| The Mean | The mean of a set of data is the sum of all the values divided by the number of values.
- Ex
x=-----
n | |
| Ex | just means the sum of all the x’s - for instance, add all the bits of data together. | |
| grouped frequency table->find MEAN | We can however, find an estimate of the mean by assuming each footballer is the height halfway within his interval | |
| (f) | means frequency | |
| "o- "definition | standart deviation->gives a measure of how the data is dispersed about the mean->the lower the standard deviation, the more compact our data is around the mean | |
| Formula standart deviation | square root of
((the sum of x2 - ((mean of x)squared))
divided by the number of units | |
| "o- 2" definition | The variance is the square of the standard deviation. | |
| The variance | is the square of the standard deviation. | |
| m-and-leaf diagram | presenting it in an easy and quick way to help spot patterns in the spread of data->They are best used when we have a relatively small set of data and want to find the median or quartiles | |
| Box-and-whisker plots (or boxplots) | These are very basic diagrams used to highlight the quartiles and median to give a quick and clear way of presenting the spread of the data. | 1.The ‘box’ part is drawn from the lower quartile to the upper quartile. The median is then drawn within the box.
2.The ‘box’ shows the inter-quartile range, which houses the middle half of the data.
3.The ‘whiskers’ are then drawn to the lowest and hig |
| Negatively skewed distribution: | There is a greater proportion of the data at the upper end. | (blank) |
| Positively skewed distribution: | There is a greater proportion of the data at the lower end. | (blank) |
| Outliers | Values of data are usually labelled as outliers if they are more than 1.5 times of the inter-quartile range from either quartile. | (blank) |
| Histograms | Histograms are best used for large sets of data, especially when the data has been grouped into classes. They look a little similar to bar charts or frequency diagrams. ->In histograms, the frequency of the data is shown by the area of the bars and not ju | (blank) |
| frequency density | The vertical axis of a histogram is labelled | frequency / class with |
| Cumulative frequency | is kind of like a running total. We add each frequency to the ones before to get an ‘at least’ total. | (blank) |
| cumulative frequency curve. | cumulative frequencies (‘at least’ totals) are plotted against the upper class boundaries to give us a cumulative frequency curve. | (blank) |
| P(A) | The probability that an event, A, will happen is written as | (blank) |
| complement of A | The probability that the event A, does not happen is called the complement of A and is written as A' | (blank) |
| mutually exclusive | Two events are mutually exclusive if the event of one happening excludes the other from happening->they both cannot happen simultaneously->When a fair die is rolled find the probability of rolling a 4 or a 1.
P(4 u 1) = P(4) + P(1)=>1/6 +1/6=>1/3 | Exclusive events will involve the words ‘or’, ‘either’ or something which implies ‘or’.->Remember ‘OR’ means ‘add’.
P(A or B) = P(A) + P(B)
P(A u B) = P(A) + P(B)
P(A u B u C) = P(A) + P(B) + P(C) |
| Independent Events | Two events are independent if the occurrence of one happening does not affect the occurrence of the other.->P(A and B) = P(A) ' P(B)
->P(A n B) = P(A) ' P(B)
Independent events will involve ‘and’, ‘both’,"either"->means multiply | A coin is flipped at the same time as a dice is rolled. Find the probability of obtaining a head and a 5.->P(H n 5)=P(H)'P(5)=> 1/2 x 1/6=> 1/12 |
| How do you write Find the probability that given he falls P(F) it was a rainy day P(R). | P(R I F) | P(R n F) / P(F) |
| discrete random variable | A random variable is a variable which takes numerical values and whose value depends on the outcome of an experiment. It is discrete if it can only take certain values. | Capital letters are used to denote the random variables, whereas lower case letters are used to denote the values that can be obtained. |
| random variable | is a variable which takes numerical values and whose value depends on the outcome of an experiment | (blank) |
| exclusive events Rewrite -> Sum? | E P(X = x) = 1 -> always sum to 1 | (blank) |
| Probability density function | Sometimes we are given a formula to calculate probabilities. We call this the probability density function of X or the p.d.f. of X. | (blank) |
| Cumulative distribution function | ‘Cumulative’ gives us a kind of running total so a cumulative distribution function gives us a running total of probabilities within our probability table. The cumulative distribution function, F(x) of X is defined as: F(x) = P(X < x) | (blank) |
| Expectation | The expectation is the expected value of X, written as E(X) or sometimes as u->The expectation is what you would expect to get if you were to carry out the experiment a large number of times and calculate the ‘mean’.. | E(X) = € xP(X = x) -> You multiply each value of x with its corresponding probability. If we then add all these up we obtain the expectation of X. This is best seen in an example. |
| uniform distribution | This is a ‘special’ discrete random variable as all the probabilities are the same.->it is possible to calculate the expectation by using the symmetry of the table. The expectation, E(X) is calculated by finding the halfway point. | (blank) |
| symmetry of the table | With uniform distributions it is possible to calculate the expectation by using the symmetry of the table. The expectation, E(X) is calculated by finding the halfway point. | (blank) |
| Expectation of any function of x | E[f(x)] = € f(x)P(X = x) | (blank) |
| E(aX + b) Equals | aE(X) + b | (blank) |
| E(a) Equals | a | (blank) |
| variance | is a measure of how spread out the values of X would be if the experiment leading to X were repeated a number of times. | (blank) |
| E(X) | -> mean -> u -> Example of Calculation->(0 x 0.1) + (1 x 0.2) + (2 x 0.5) + (3 x 0.2) | (blank) |
| Var(aX) Equals | a2Var(X) | Var(2X) = 22 x Var(X)
= 4 x 2.5 = 10
Var(4X – 3) = 42 x Var(X)
= 16 x 2.5 = 40
|
| Var(aX + b) Equals | a2Var(X)
This means by knowing just the variance, Var(X), we can calculate other variances quickly.
Example: | Var(2X) = 22 x Var(X)
= 4 x 2.5 = 10
Var(4X – 3) = 42 x Var(X)
= 16 x 2.5 = 40
|
| The Standard Deviation | The square root of the Variance is called the Standard Deviation of X. standard deviation is given the symbol o- | (blank) |
| convert any normal distribution of X into the normal distribution of Z | (X - u) / o- | (blank) |
| Normal Distribution Graph | much of the data is gathered around the mean. The distribution has a characteristic ‘bell shape’ symmetrical about the mean. ->The area of the bell shape = 1. | (blank) |
| The standard deviation | is an important measure of the spread of our data. The greater the standard deviation, the greater our spread of data. | (blank) |
| § | this Greek letter just describes the area under the bell from that point! | (blank) |
| line of best fit’ | Any line of best fit must go through the mean of x, and the mean of y. | (blank) |
| linear correlation | If all (or nearly all) of these points seem to lie in a straight line | (blank) |
| Equation of regression line | (blank) | (blank) |
| Regression Line x on y->Formula for b: | Sxy / Syy | (blank) |
| Regression Line y on x->Formula for b: | Sxy / Sxx | (blank) |
| Independent/dependent variables | With the above data, x looks to be controlled, where y appears to be dependent on an experiment and x. In this case, we say that x is an independent variable and y a dependent variable. As x appears controlled and accurate we only need to calculate the re | (blank) |
| product moment correlation coefficient | r -> is a measure of the degree of scatter.->will lie between -1 and 1. | (blank) |
| "r" | The product moment correlation coefficient, r, is a measure of the degree of scatter.->will lie between -1 and 1. | (blank) |
| Calculate E(X) | €x times P(X = x) / or € f(x)P(X = x) | (blank) |
Ch.18 LTI-MA 509
