My aim is that within the limits of a small-scale survey I will collect sample data of a population, and by using estimation techniques I will determine the population's parameters (such as the mean and the variance).

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Mathematics Coursework – Statistics.

S1 Task A: Measurements.

Aim.

My aim is that within the limits of a small-scale survey I will collect sample data of a population, and by using estimation techniques I will determine the population’s parameters (such as the mean and the variance). My population is smarties, and in this investigation I am looking at the individual weight of random smarties, which will be my sample. I decided to stick with weight, as it is a property that will vary a lot, I think, and so I hope will prove an interesting investigation. An important factor to help me decide on how large my sample should be is that the size of the sample must be quite small, because it is stated so in my aim. However, to make accurate estimates of population parameters the sample must be large enough.

Therefore to help me decide on the size of my sample, I have accordingly looked at the Central Limit Theorem, which states that:

  • If the sample size is large enough, the distribution of the sample mean is approximately Normal.
  • The variance of the distribution of the sample mean is equal to the variance of the sample mean divided by the sample size.

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 fifty, 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”.

The sample.

The sample will be of the weight of fifty smarties. To be a “good” sample I must make sure that the results are valid and not biased in any way, which means that these smarties must be collected randomly, because the sample must be random for the Central Limit Theorem to be in effect, which would provide a Normal distribution of its mean which will allow me to make predictions of the parent population.

I have decided to collect my sample data in a group, in order to lower he costs of the investigation, and also to provide assistance with greater accuracy to collecting the sample. Five tubes of smarties will be bought, each from a different shop, and ten smarties will be selected at random from each tube to be used in the survey. This should produce a nice sample, which I am fairly confident will be random.


Calculations.

In this investigation I will need to calculate a few things in order to come to my conclusion. This is a quick plan of what I will be calculating.

  • The mean, standard deviation and variance of the sample.
  • These will be used to estimate the variance and standard deviation of the parent population of smarties.
  • This in turn, will be used to estimate the standard error (the standard deviation of the sample mean distribution).
  • And, this will be used along with the mean of the sample to create confidence intervals for the mean of the parent population of smarties.
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Accuracy of measurements.

My group will randomly select the required smarties from the packets and one after the other, they will be weighed on an electronic balance that will be “reset” to zero after each measurement, which will reduce the chance of any inaccuracies that might arise from small pieces of smartie being left on the balance.

The balance we have used has a high degree of accuracy, as the measurements are given in grams to three decimal places. However, if the difference in the weight of smarties is too small to be detected on this balance, either a ...

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