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Undertake a small-scale survey to estimate population parameters.

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Introduction

Undertake a small-scale survey to estimate population parameters.

Aim

Undertake a small-scale survey to estimate population parameters.

Size of Sample

The size of the sample must be quite small, because it is stated so in the aim. However, to make accurate estimates of population parameters the sample must be large enough.

According to the Central Limit Theorem:

n If the sample size is large enough, the distribution of the sample mean is approximately Normal.

n The variance of the distribution of the sample mean is equal to the variance of the sample mean divided by the sample size.

These are true whatever the distribution of the parent population. The Central Limit Theorem allows predictions to be made about the distribution of the sample mean without any knowledge of the distribution of the parent population, as long as the sample is large enough.

For this reason, the sample size will be set at 50, which I consider large enough for the distribution of its mean to be normal (according to the Central Limit Theorem). It should not be larger because the aim of this investigation is to carry out a “small scale survey”

How / What Data to be Collected

The sample will be of the weight of 50 smarties. To be a “good” sample, that is that the results are valid and not biased in any way, these smarties must be collected randomly.

Middle

1.027

0.922

1.110

0.932

0.955

1.050

0.934

0.972

1.011

0.901

1.045

0.893

1.034

0.955

0.957

1.047

0.934

1.041

0.959

1.081

0.915

Mean, Standard Deviation and Variance of Sample

Estimate of the Variance of the Population of Smarties

The variance of the sample is a biased estimator. A biased estimator is one for which the mean of its distribution is not equal to the population value it is estimating. To convert the variance of the sample to an unbiased estimator it must be multiplied by where n is the size of the sample.

This figure can then be used to estimate the variance of the parent population.

Standard Error

The standard error is the standard deviation of the sample mean. According to the central limit theorem, the variance of the sample mean can be calculated by dividing the variance of the population (estimated above) by the size of the sample. The standard error can be calculated by performing a square root of the variance of the mean. This can be demonstrated algebraically:

Estimate of the Mean of the Parent Population

The mean is an unbiased estimator, that is, the mean of its distribution is equal to the mean of the parent population. For this reason it can be used as an estimator for the mean of the population of smarties. An estimate of the mean of the population of smarties is therefore 0.976.

The standard error calculated above is quite small. This means that the variance of the sample mean is low, and this shows that one can be quite confident that the actual mean of the population is around 0.976. However this is not a very “mathematical” or “user friendly” method of showing how confident one is about the accuracy of the estimate made.

Confidence Intervals Background

To calculate how confident one is about the estimate of the population mean, one can use confidence intervals. These tell you how confident (as a percentage) you can be that the mean of the population falls within a given range. How they work is explained in the following.

Conclusion

The sample might have been a “fluke” I might have got all the big smarties, or all the small ones. However there is not much to do to eliminate the possibility of this apart from to weigh every single smartie. This is extremely impractical (possibly impossible).

The smarties gathered were from my immediate area. Even though they were taken from different shops and different packets, they do not necessarily represent all the smarties in the world, only ones in my area.

The results may be unreliable because the company that produces smarties may be changing, or have changed the mean weight setting for the smarties. They may be trying to slowly lower the weight while keeping the price the same. This could mean that the actual population parameters are somewhat different to the ones estimated here.

Possible Extension

A statistical analysis of entire tubes of smarties could be carried out. The actual weight of the smarties could be compared to the price on the tube to determine whether the manufacturers are lying about how much smartie there is in their packets.

Weighing smarties of different colours could also be done to find if there are any differences between them.

Also, a larger sample size could be taken to determine the mean and variance more accurately

This student written piece of work is one of many that can be found in our AS and A Level Probability & Statistics section.

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