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Chapter 5   Sampling and Variability

5.1   Probability and distributions

5.1.1   Equally likely outcomes

This exercise asks for probabilities about a card drawn at random from a standard pack of playing cards or when two dice are rolled.

5.1.2   Value from finite categorical population

A finite categorical population is shown as a bar chart and the probability of a randomly sampled value being in a range of categories must be found.

5.1.3   Value from finite numerical population

This exercise is simlar to the one on the previous page but relates to a numerical population that is summarised by a histogram. The probability must be estimated roughly 'by eye' from the area under the histogram.

5.1.4   Random value from infinite population

In this exercise, probabilities are obtained from the areas under a probability density function.

5.1.5   Equally likely outcomes

This exercise relates to pairs of numbers drawn with or without replacement from a small finite population. The question asks for a probability about either the sum or difference of the two numbers.

5.2   Normal distributions

5.2.1   Shape of normal distribution

One exercise asks you to guess the standard deviation of a normal distribution. Another requests a sketch of a normal distribution given its mean and standard deviation.

5.2.2   Probability and area under normal curve

Exercise asks for a normal probability by reading an area under a normal histogram (pdf).

5.2.3   Normal probabilities from z-scores

The two exercises on this page are similar to the previous exercise, but the question must be first translated into one involving a z-score. In the first exercise, the required probability is read off a standard normal histogram (pdf); in the second, a normal table must be used.

5.2.4   Expected numbers

Instead of the probability of getting a value within some range, the questions in this exercise ask for the expected number out of n items.

5.2.5   Finding X from probability

The two exercises on this page ask the inverse problem; given a probability, they ask for the corresponding range of values. The first exercise uses a standard normal histogram (pdf) to find the answer, whereas the second uses a normal table.

5.2.6   Mixture of questions

The exercise on this page asks a variety of different questions of the types above about normal distributions. The reader must pick the type of question before using the templates and normal table to answer.

5.2.7   Guessing normal probabilities

The 70-95-100 rule of thumb gives the approximate probability of a value within 1, 2 and 3 standard deviations of the mean. Te exercise uses this rule of thumb to find some normal probabilities without formal calculations.

5.3   Distribution of sample proportion

5.3.1   Recognising a binomial distribution

This exercise describes four random variables and asks which have binomial distributions.

5.3.2   Shape of binomial distribution

You are given information about the parameters of a binomial distribution in these two exercises and are asked to pick which of four alternative bar charts describes it. In the first exercise, the parameters are directly provided but the second exercise requires careful reading of the wording of the question.

5.3.3   Finding binomial probabilities

In this exercise, you will use the information in the question to set the parameters of a binomial distribution then read off the probability by dragging over the bar chart of the binomial probabilities. (The computer does the calculations.)

5.3.4   Normal approximation

You are asked to find the parameters of a normal approximation to a binomial distribution in this exercise.

5.3.5   Probabilities from normal approximation

This exercise asks for probabilities relating to a binomial distribution but requires that you use a normal approximation to evaluate them.

5.4   Distributions of means & sums

5.4.1   Spread of distribution of mean and sum

This exercise shows the distribution of the mean or sum of values from samples of different sizes from a normal population. You must match the distributions with their sample sizes.

5.4.2   Shape of distribution of mean and sum

This exercise examines understanding of the limiting normal distribution of the sample mean and sum as sample size increases, whatever the shape of the population distribution.

5.4.3   Distribution of mean and sum (cont)

In this exercise, pop-up menus are used to specify the centre, spread and skewness of the distribution of a sample mean, sample sum or a single value, given the population distribution.

5.4.4   Probabilities relating to means and sums

This exercise asks for the probability that the mean, the sum or a single value from a sample lies within some interval. Templates are provided to help evaluate the parameters of the sampling distribution and the z-scores for the endpoints of the interval.