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Acknowledgements
Representation of data
Why do we collect, display and analyse data?
Types of data
The number of letters in the words of a book is an example of discrete quantitative data. The times it takes the athletes to complete a 100-meter race are an example of continuous quantitative data.
Representation of discrete data: stem-and-leaf diagrams
Continuous data can take any value (possibly within a limited range), as shown in the diagram. Describe the student in the middle of the line and find the largest possible number of boys in the line who are not standing next to a girl.
Representation of continuous data: histograms
The following table shows the intervals between the classes and the number of animals in the two classes. Given that the frequency densities of the four classes of percentage scores are in a ratio, find the value of p and q.
Representation of continuous data: cumulative frequency graphs
9 The following table shows the age of the students at a university, expressed as a percentage. Explain why, from your graphs, you cannot get an accurate estimate of the number of these 242 sticks that are acceptable.
Comparing different data representations
Name a type of representation that shows that the majority of eggs have a mass of 54 to 60 g. 8 The following table shows the focal lengths, lmm, of the 84 zoom lenses sold by a retailer.
Checklist of learning and understanding
Given that a–b=7 and that the sum of the eight numbers correct to the nearest whole number is 24. Given that the height of the column representing the combined classes must be 30 cm, find the correct height for the.
Measures of central tendency
Three types of average
The mode and the modal class
Find the modal class of the 270 pencil lengths, given to the nearest centimeter in the following table. Find the least possible frequency of the modal class, given that the modal class is 4–10.
The mean
Combined sets of data
Means from grouped frequency tables
The following example shows a situation where using incorrect bounds leads to an incorrect estimate of the mean. Calculate an estimate of the difference between the mean heights of these two groups of children.
Coded data
The median
In a set of n ordered values, the mean is at the value halfway between 1 and n. To find the average number of customers served on each of these days, we need to find their positions in the ordered rows of the back-to-back stem-and-leaf diagram.
Estimating the median
There are no historical examples of the use of the mean, median, or mode before the 17th century. He used the number in the middle of the smallest and largest values (what we would call the midrange) and ignored everything but the minimum and maximum values.
Choosing an appropriate average
Investigate the effect this has on the mode, mean, median and shape of the curve. What the shape says about the mean, median, and mode values.
Measures of variation
How do we best summarise a set of data?
The range
The interquartile range and percentiles
Ungrouped data
Find the interquartile range of the 13 pooled values shown in the following stem-and-leaf plot.
Grouped data
Variance and standard deviation
Find the standard deviation for the values of x given in the following table, correct to 3 significant figures. Calculate an estimate of the standard deviation of the heights of the 20 children listed in the following table. Some features of the standard deviation are compared with the interquartile range in the following table.
Calculating from totals
Correct to 1 decimal place, the standard deviation of the number of pages in the 15 books together is 31.2. And how can we find the variance and standard deviation of the original data from the coded data. Use the coded values to calculate the standard deviation of the number of brothers, to 3 decimal places.
Appendix to Section 3.3
Probability
If we do this, how likely is that?
Experiments, events and outcomes
Random selection and equiprobable events
The probability of an event occurring is equal to the proportion of equally likely outcomes that are favorable to the event. There are 19 possible outcomes: 11 are favorable for the meeting to choose a boy and eight are favorable for the meeting to choose a girl, as shown in the following table.
Exhaustive events
Trials and expectation
Mutually exclusive events and the addition law
Mutually exclusive events have no common favorable outcomes, meaning that it is not possible for both events to occur, so P(Ain )B =0. For example, when rolling an ordinary die, the event even and the factor are mutually exclusive because they do not share favorable outcomes. Events are not mutually exclusive if they have at least one favorable outcome in common, which means that both events can occur, i.e. P(Ain )B ≠0.
Venn diagrams
Independent events and the multiplication law
Find the probability that the sum of the points of three tosses of a regular fair die is less than 5. 1 Using a tree diagram, find the probability that exactly one head is obtained when two fair coins are tossed. Evaluate k and find the probability that the sum of the three points is less than 5.
Application of the multiplication law
Conditional probability
9 The additional information 'given that a girl is chosen' reduces the number of possible choices from 20 to 9 and Rose is one of those nine girls. Given that the sum of the two numbers rolled is even, find the probability that the two numbers are the same. Given that a player's score is greater than 6, find the probability that it is not greater than 8.
Independence and conditional probability
Dependent events and conditional probability
Given that he is late for school, find the probability that he rides a bicycle; that is, find P(C L|. Find the probability that the number is a multiple of 5, given that none of its digits is a 5. Find the probability that three randomly selected members have all read fewer than eight books , given that.
Permutations and combinations
Simple situations with millions of possibilities
The factorial function
A shorthand method of doing this is to use the factorial function, called 'quadrifactorial' and written 4.
Permutations
We can show that six three-digit numbers can be made from 5, 6, and 7 by considering how many choices we have for the digit we put in each position in the arrangement. Finally, we place the remaining digit on the right side, as shown in the following diagram. If her youngest child sits on the adjacent chair to her left, in how many ways can the remaining children sit.
Find the number of ways in which five cars and x+2 vans can be parked in a row. If two more boys are added to the group, the number of possible arrangements increases by a factor of 420. 4 Two students should find out how many ways they can plant two trees and three bushes in a row.
Alternatively, the shortest man can be placed in one of the four positions, and the remaining five positions can be filled in 5P5 fashion, i.e. 4 × 5. The remaining four positions can be filled by any of the other four men (one of whom one is the shortest man) in 4P4 fashion, as shown. Start with 3: We need to place 1 on the far right (one choice), and the remaining two positions can be filled in 2P2 fashion by the other two numbers, as shown.
Combinations
How many different three-digit numbers can be made from five cards, each with one of the digits and 9 written on it. 6 From six boys and seven girls, find how many ways there are to select a group of three children that consists of more girls than boys. In how many ways can she arrange seven of the objects in a row along the shelf if her clock is to be included.
Problem solving with permutations and combinations
3 Find the probability that the arrangement of all the letters in the word PALETTE will be chosen at random. Find the probability that the first card in a row is odd and the three cards are in the middle. Therefore, find the probability that this event does not occur in two consecutive throws of three dice.
Probability distributions
2 Two dice are randomly selected from a bag of three red dice and three blue dice. Show that the chosen dice are more likely to be red when the choices are made with replacement than when the choices are made without replacement.
Tools of the trade
Discrete random variables
Probability distributions
Three DVDs are selected and the following table shows the probability distribution for M, the number of movies selected. Set up the probability distribution for X, the number of right-handed, red-haired people selected, and state what assumption must be made to do so. The probability distribution for H, the number of heads obtained, is shown in the following table.
Expectation and variance of a discrete random variable
Expectation
From this table of expected frequencies, we can calculate the mean (expected) score over 1600 trials. The same value for E(X) is obtained if we use relative frequencies (i.e. probabilities) instead of frequencies.
Variance
A probability distribution for a discrete random variable is a representation of all its possible values and their respective probabilities. 1 Find the mean and variance of the discrete random variable X whose probability distribution is given by. Find the probability that the result is at least 4 in at least 1 of the 3 tosses.
The binomial and geometric distributions
Two special discrete distributions
The binomial distribution
Find the probability that less than 39 people in a random sample of 40 have rhesus positive blood. Find the probability that exactly five out of eight randomly selected people succeed on their first attempt. Find the probability that there is exactly one person in the sample who is color blind.
Expectation and variance of the binomial distribution
The geometric distribution
The following table shows the probability that the first success occurs on the rth trial. A random variable X having a geometric distribution is denoted by X~ Geo( )p, and the probability that the first success occurs on the rth trial is X is the number of candies selected and eaten, up to and including the first red candies.
Mode of the geometric distribution
Expectation of the geometric distribution
4 Let T be the number of times a fair coin is tossed, up to and including the toss in which the first tail is obtained. 5 Let X be the number of times an ordinary mushroom is rolled, up to and including the roll in which the first 6 is obtained. X is the number of randomly selected people, up to and including the first person who has this defective gene.
The normal distribution
Why are errors quite normal?
Continuous random variables
