14 1popsandsamples

# Population Parameters and Sample Statistics

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This topic involves the study of taking samples from a population and then using data from the sample to draw conclusions about the population.
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Polulation Sample
A population is the entire group that you are interested in. A sample is a small group chosen from the population.
The population may be very large in which case it may be impractical to observe/question/measure each member of the population. By observing/questioning/measuring the sample, we can draw conclusions about the entire population.
N represents the size of the population (sometimes the exact value of N is unknown) n represents the size of the sample
A parameter is a characteristic of the entire population. A statistic is a characteristic of a sample taken from the 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. 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 parameter is assumed to be constant The value of a statistic will vary from sample to sample

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## Binomial Data

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

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Population Sample
The population proportion is the proportion of the entire population who score yes for the question being studied. The sample proportion is the proportion of the sample who score yes for the question being studied.
p represents the population proportion $\hat{p}$ represents the sample proportion
p is a parameter and is assumed to be a constant. $\hat{p}$ is a statistic and will vary from sample to sample
$p = \dfrac{\text{num of successes in population}}{\text{population size (N)}}$ $\hat{p} = \dfrac{\text{num of successes in sample}}{\text{sample size (n)}}$

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

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## Sampling Techniques

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A good sample should be representative of the population.
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### 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.
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### Example 1

A researcher wishes to evaluate how well the local library is catering to the needs of a town’s residents. To do this she hands out a questionnaire to each person entering the library over the course of a week. Will this method result in a random sample of the town? What is a better method?

Solution

• The sample would only contain people who use the library so this is not random.
• The researcher should select people randomly at different locations around the town

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### Systematic Sample

… This assumes the members of the population are randomised and then placed in some sort of order

… Every k member of the population is chosen for the sample. {where k is some integer|

… Eg every 10th member is selected
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### Example 2

A production line produces chocolate bars. Every 15th chocolate bar is selected and tested to ensure it passes quality control. Will this method result in a random sample? What assumption has been made?

Solution

• For most situations, this would produce a random sample.
• The assumption is that there is no pattern to the way chocolate bars end up on the production line.
• For example, if the chocolate bars came out of a mold with 15 slots, then every 15th bar might have come out of the same slot in the mold.

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### Stratified Sample

… This method is appropriate when there are subgroups within the population

… The subgroups 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
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### Example 3

A sample of 10 students is to be selected from a group of 200 students, of whom 123 are in Year 11 and 77 are in Year 12.

How many students should be selected from each Year 11 and Year 12 according to Stratified Sampling.

Solution

• The proportion of Year 11s in the population is $\dfrac{123}{200} = 0.615$

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• So the number of Year 11s for the sample is:
• $0.615 \times 10 = 6.15$
• so 6 Year 11 students

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• So the number of Year 12s for the sample is:
• $10 - 6 = 4$
• so 4 Year 12 students

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### Self-Selected Sample

… participants volunteer to be in the sample

… samples obtained using this method are almost never representative of the total population

… phone-in polls or online competition voting are examples of this style of sampling

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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

• Press KEYBOARD
• Tap down arrow on bottom left of keyboard screen
• In CATALOG, select RANDLIST(
• Type: 10, 1, 30)
• Press EXE

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

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