• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

Undertake a small-scale survey to estimate population parameters.

Extracts from this document...


Undertake a small-scale survey to estimate population parameters.


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.

...read more.























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.

...read more.


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

...read more.

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

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related AS and A Level Probability & Statistics essays

  1. The mathematical genii apply their Statistical Wizardry to Basketball

    = 0.2065493847 x 80 = 16.7161869 Let Y be the number of attempts taken before a basket is scored for Dom: Probability of scoring a basket: P(score) = sample size/total number of shots = 80/345 = 0.231884058 This implies Y~G ( 0.232 )

  2. Standard addition was used to accurately quantify for quinine in an unknown urine sample ...

    The sample beam first passes through an excitation filter or a monochromator, which transmits radiation to excite fluorescence but exclude radiation of the wavelength of the fluorescence emission. Fluorescence from the sample is propagated in all directions but is most conveniently observed at right angles to the excitation beam; at

  1. I am investigating how well people estimate the length of a line and the ...

    Hypothesis 1 (Year 10 will be better at estimating the size of an acute angle than Year 7): * I will sort (in ascending order) Year 7 Males and Females combined, to use when drawing out the cumulative frequency table.

  2. Guestimate - investigate how well people estimate the length of lines and the size ...

    Also, to make the comparison between year 10 and year 7 easier to read I have put both the lines for each angle or line on the same graph. I also worked out the inter-quartile range. I will do a histogram for the angle or line that I thought needed the most explaining.

  1. "The lengths of lines are easier to guess than angles. Also, that year 11's ...

    43 1 17.5 -16.5 272.25 4 50 4 23.5 -19.5 380.25 4 45 4 20.5 -16.5 272.25 5 30 17.5 3 14.5 210.25 5 35 17.5 8 9.5 90.25 5 45 17.5 20.5 -3 9 TOTALS -3.5 3494.25 Now, to find the Spearman's coefficient of rank you have to use the formula: p = 1 - 6?d� n(n� - 1)

  2. Chebyshevs Theorem and The Empirical Rule

    For instance, for k = 2.5 we get the result that in the interval years Example 1: Students Who Care is a student volunteer program in which college students donate work time in community centers for homeless people. Professor Gill is the faculty sponsor for this student volunteer program.

  1. Data Analysis of American House Price

    In this case it worth to continue investigating the relationship. However, the point are scattered quite broadly, so it is necessary to analyse the value of r in order to determine how strong the relationship is. The correlation coefficient (0,65)

  2. Used Cars - What main factor that affects the price of a second hand ...

    The line of best shows that the value of a car will go into the minuses at the age of 10, but this can't be true because there is a point on the scatter graph that shows that there is a car which is 11 years old and has a value around �2000.

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work