A142popsandsamples


 * Population Parameters and Sample Statistics **

A ** population ** is the entire group that you are interested in.
 * The population may be very large in which case it may be impractical to observe/question/measure each member of the population.
 * We use ** N ** to represent the size of the population (sometimes the exact value of N is unknown)

A ** sample ** is a small group chosen from the population.
 * By observing/questioning/measuring the sample, we can draw conclusions about the entire population.
 * We use ** n ** to represent the size of the sample

A ** parameter ** is a characteristic of the entire population.
 * For example if we use all the students at the school on one day as the population,
 * then the mean and standard deviation of the heights of all students at the school are both parameters.
 * The value of a parameter is assumed to be constant

A ** statistic ** is a characteristic of a sample taken from the population.
 * In the example of students at the school, if we took a sample of 20 students
 * then the mean and standard deviation of the heights of students in the sample are both statistics.
 * The value of a statistic will vary from sample to sample

For this unit, we are restricting our study to binomial data, so each data point will be either yes/no (or success/fail).

The ** population proportion ** is the proportion of the entire population who score yes for the question being studied.
 * We use ** p ** to represent the population proportion
 * p is a parameter.
 * p is assumed to be a fixed value (constant)

math . \qquad p = \dfrac{\text{number of successes in population}}{\text{population size (N)}} \qquad. math

The ** sample proportion ** is the proportion of the sample who score yes for the question being studied. math . \quad \bullet \quad \text{We use } \hat{p} \text{ to represent the sample proportion } \qquad. \\ . \quad \bullet \quad \hat{p} \text{ is a statistic} \\ . \quad \bullet \quad \hat{p} \text{ will vary from sample to sample} math

math . \qquad \hat{p} = \dfrac{\text{number of successes in sample}}{\text{sample size (n)}} \qquad. math

In this unit we will use a study of the sample proportion to draw conclusions about the population proportion.


 * Sampling Techniques **

A good sample should be representative of the population.


 * Random Sample **
 * A randomisation method is used to identify the members to be included in the sample
 * Each member of the population has the same probability of being selected for the sample
 * A common method is to use a computer/calculator to generate a list of random numbers where each number corresponds to a member of the population.


 * Systematic Sample **
 * This assumes the members of the population are randomised and then placed in some sort of order
 * Every kth member of the population is chosen for the sample.
 * Eg every 10th member is selected


 * Stratified Sample **
 * This method is appropriate when there are subgroups within the population which may have a differing likelihood of scoring a success
 * A random sample is taken from each subgroup in proportion to the size of that subgroup

A sample of 10 students is to be selected from a group of 200 students, of whom 113 are in Year 11 and 77 are in Year 12.
 * Example**

math . \qquad \text{The proportion of Year 11s in the population is } \dfrac{123}{200} = 0.615 \qquad. \\ . \qquad \text{So the number of Year 11s in the sample is: } 0.615*10 = 6.15 \; \text{ so 6 Year 11 students} \qquad. \\ . \\ . \qquad \text{So the number of Year 12s in the sample is: } 10 - 6 = 4 math


 * Self-Selected Sample **
 * participants volunteer to be in the sample
 * the sample obtained using this method is almost never representative of the total population
 * phone-in polls or online competition voting are examples of this style of sampling


 * Using the Casio Classpad **
 * We can use the Classpad to produce a list of random numbers
 * This list can then be used to select a random sample from a population


 * 1) Press KEYBOARD
 * 2) Tap down arrow on bottom left of keyboard screen
 * 3) In CATALOG, select RANDLIST(
 * 4) Type 10, 1, 20)
 * 5) Press EXE

RANDLIST(10, 1, 30) will produce a list of 10 random numbers between 1 and 30

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