The aim of this investigation is to collect data from a population and using the results estimate population parameters i.e. mean. To collect my data I am required to measure one factor of the chosen population, these factors include:* Heights* Weights* Pulse rates* Reaction times of males and females* Age group’s etc.I have decided that my population will be two types of sweets and the factor that I am going to measure is the weights of each individual sweet. The two types of sweets I have decided to use are:* Galaxy Minstrels* MaltesersWhen collecting my data I will have select an appropriate method to obtain the information that I require. There are two methods in which data can be collected:* Survey – used to determine some particular characteristic(s) of a population, usuallydone through questionnaires.* Experiment – used to test a factor when that factor is the only variable.The most suitable method to use to measure the weights of individual sweets is by doing an experiment.It would be impossible to collect the weights of each sweet in the whole population a sample of the population must be taken. A suitable sample size must be chosen which will allow me to obtain reliable data in which I could draw a conclusion from. Hence, I will be using a sample size 100 of each type of sweet, as this size will be appropriate for my requirements.To obtain a sample of 100 sweets from the whole population it is necessary to choose a practical sampling procedure. For this investigation there are two main methods that could be used:* Stratified sampling – This is takes the same proportion of each population group and obtaining data from that proportion.* Random sampling – This sampling method is where every member of the population has an equal chance of being selected. Hence, each member is selected completely at random.I will be taking a random sample. I will obtain my random sample by buying a couple of bags of Galaxy minstrels and Maltesers on different dates i.e. a few days apart, next I will mix all the Galaxy minstrels together and place them in a bag (I will do the same to the Maltesers). To select a sample of 100 sweets I will randomly pick out 100 sweets from each of the bags and weigh them individually.Once I have obtained my data I will use it to calculate the mean and variance weights of the sweets and the standard error, using the standard error I am hoping to be able to calculate the confidence intervals of the Galaxy minstrels and the Maltesers. From this I will be able to draw suitable comparisons and a conclusion of the sweets.MethodTo obtain my data I will be doing an experiment; I will firstly take a random sample of 100 Galaxy minstrels and 100 Maltesers (as described above). I will the measure the weights of each Galaxy minstrel and Maltesers individually using weighing scales (accurate to 2 d.p) and make a record the weights.ResultsAfter doing this experiment I have obtained the following weights of the sweets:Galaxy minstrels (g):2.442.732.562.672.562.572.672.612.472.642.612.482.712.532.642.722.692.712.732.532.692.582.512.632.702.562.662.542.652.562.762.552.572.432.662.492.642.652.672.472.692.612.682.722.452.642.622.642.812.672.682.732.782.672.732.562.632.482.472.722.682.672.422.612.552.652.472.662.522.682.502.672.722.552.572.512.612.532.692.642.632.652.682.542.532.482.492.492.532.742.682.482.682.652.622.752.722.772.702.48Maltesers (g):1.882.031.981.482.162.132.091.821.852.111.782.322.172.932.102.052.512.451.761.601.602.241.732.432.651.702.101.892.181.621.442.032.182.201.881.641.952.251.871.712.111.952.272.082.381.872.272.022.151.431.692.221.702.461.482.152.351.891.691.162.192.042.662.012.332.092.312.271.862.231.891.972.762.042.142.332.151.731.341.671.741.921.762.412.131.491.801.961.912.552.492.511.852.061.402.022.072.092.162.18Mean, Variance and Standard deviationNow that I have obtained all the weights of each individual sweet I will be able to calculate the mean, variance and standard deviation of the sample.Mean (u)The mean, u, is the sum of all the data values divided by the number of observations, it is a measure of the ‘average’ value of a set of data. The mean is equally influenced equally by all the data values. The formula for calculating the mean is:u = ?xnI will calculate the mean of the Galaxy minstrel’s and Maltesers separately.Galaxy Minstrels u = ?x = 261.41 = 2.61g (2 d.p)n 100Maltesers, u = ?x = 202.32 = 2.02g (2 d.p)n 100VarianceThe variance, o ï¿½, is calculated by taking the mean of the sum of the squared deviations from the mean. The formula to calculate variance is:o ï¿½ = ?x ï¿½ – x ï¿½nI will once again calculate the variance of the Galaxy minstrels and Maltesers separately.Galaxy Minstrels, o ï¿½ = ?x ï¿½ – x ï¿½ = 684.2055 – 2.61ï¿½n 100= 6.842 – 6.812=0.03 (2 d.p)Maltesers, o ï¿½ = ?x ï¿½ – x ï¿½ = 419.735 – 2.02ï¿½n 100= 4.197 – 4.080= 0.12 (2 d.p)Standard DeviationThe standard deviation, o , is the measure of spread around the mean. It is calculated using this formula:o = /?x ï¿½ – x ï¿½nAs you can see the standard deviation is simply the squared root of the variance:VvarianceAs I have already calculated the variance it will be relatively easy to calculate the standard deviation (by square rooting the variance)Galaxy Minstrels o = Vvariance = V0.03= 0.17 (2 d.p)Maltesers o = Vvariance = V0.12= 0.35 (2 d.p)Now that I have calculated the mean, variance and standard deviations of the Galaxy minstrels and Maltesers I will be able to compare them to do this I will place the results in a table as it will be easier to look at the data:MeanVarianceStandard DeviationGalaxy Minstrels2.610.030.17Maltesers2.020.120.35From the table you can see that the ‘average’ weight (or mean) of the Galaxy minstrels is higher than the mean weight of the Maltesers, this general shows that the Galaxy minstrels weigh more than the Maltesers.The weights of the Galaxy Minstrels are more regular than the Maltesers this is shown by both the standard deviation and the variance. The variance and standard deviation of the Maltesers is a lot higher than the Galaxy minstrels. There would be a greater confidence in predicting the actual weight of a galaxy minstrel than the weight of a maltesers as the weights of the Galaxy minstrels are more tightly bunched together and therefore has less variation.The weights of the Galaxy minstrels are 2.61g + 0.17g and the weight of Maltesers is 2.02g + 0.35g. Hence Galaxy minstrels generally weigh more.The Normal DistributionNormal distribution curves can be used as a model for the distribution of a number of continuous variables. Most distributions that occur are bell shaped, but before an accurate normal distribution curve can be drawn the data must be standardised.The Standard Normal distribution has a curve as shown below:The standard normal distribution has a mean of 0 and a standard deviation 1, hence the area under the curve is 1.The area under the standard normal curve to the left of a point (e.g. z) is denoted by (z):The shaded area is would be calculated using the normal distribution tables.Standardising dataAs stated before to draw a suitable normal distribution curve the data being used must be standardised. This is calculated by using the formula:Z = x – xsWhere x is the mean, x is the observation and s is the standard deviation.e.g. The standardised form of the first weight of a Galaxy minstrel is:Z = x – x = 2.44 – 2.61s 0.17= – 0.170.17= -1This means that the weight 2.44g is 1 standard deviation units below the mean.Normal Distribution CurveWhen looking at the two sweets; Galaxy minstrels and Maltesers. I think that the normal distributions will be normal (bell-shaped) as these sweets are manufactured goods, hence, made by machines therefore the weights of each sweet will be very similar to each other and not have huge variations between them. So when a normal distribution curve is drawn it will be normal.After standardising the data (as shown above) I was able to draw a normal distribution curve:Galaxy Minstrels:As you can see from the curve above the distribution of the galaxy minstrels is normal. I will now draw a normal distribution curve for the Maltesers:The distribution of the Maltesers is also normal.To compare the distributions of the two sweets I will place the two curves on the same axis, as shown below:When the two curves are placed together it is obvious that the standardised weights of the Maltesers are more clustered around the mean than the Galaxy minstrels.The Central Limit TheoremThe central limit theorem states that the mean of the sample is the same as the mean of parent population, it also states that the sample variance is the variance of the parent population divided by the sample size (as stated above).The central limit theorem is crucial to work on sampling; it allows you to make predictions about the distribution of the sample without necessary knowing the distribution of the parent population.An unbiased estimator is when the mean of its distribution is equal to the population value that it is estimating. Hence, the sample mean is an unbiased estimator of the population mean.Population parametersWhen looking at populations there are two types of populations to consider:* Parent population – this is the whole population e.g. all the galaxy minstrels ever made.* Sample population (sample means) – this is a small amount of the population.These can be illustrated as:The distribution of the sample means is normal; the mean of the distribution of the sample means is equal to the mean of the parent population, but the results for the sample are more clustered around it’s mean than the parent population this demonstrates that the sample has a small variance than the parent population.Hence, mean, u, is the same for both and if the sample size, n, is given the variance will be o ï¿½nStandard ErrorThe confidence of calculating the weight of a sweet is based on the variation of the weights, if there is less variation the confidence increases. This suggests that you might use the variance of the sample mean values as a measure of confidence.When estimating the size of the sample and its standard deviation. The standard deviation of the mean is called the standard error. The standard error is calculated using the formula:s.e. = oVnUsing this formula I will calculate the standard error for both the Galaxy minstrels and the Maltesers.Galaxy minstrel s.e. = o = 0.17 = 0.017 (2 s.f)Vn V100Maltesers s.e. = o = 0.35 = 0.035 (2 s.f)Vn V100Confidence IntervalsConfidence intervals are used as a ‘user-friendly’ way to express how confident you are in your estimate of a population mean. You can use the sample data to construct an interval estimate for the population mean.Confidence intervals are calculated by using the standard error, mean and the normal distribution tables. This is shown in the formula:% confidence interval = (sample mean – N.D.V * s.e, sample mean + N.D.V * s.e)Where N.D.V is the normal distribution value.Using this formula I will firstly calculate the 70% confidence interval for each of the sweets.Galaxy Minstrels70% confidence interval = (sample mean – N.D.V * s.e, sample mean + N.D.V * s.e)= 2.61 – 0.525 * 0.017, 2.61 + 0.525 * 0.017= 2.60, 2.62This shows that there is a 70% confidence that the weight of a Galaxy minstrel weighs between 2.60g and 2.62g.Maltesers:70% confidence interval = (sample mean – N.D.V * s.e, sample mean + N.D.V * s.e)= 2.02 – 0.525 * 0.035, 2.02 + 0.525 * 0.035= 2.00g, 2.04gThis shows that there is a 70% confidence that the weight of a maltesers weighs between 2.00g and 2.04g.Now I will find the 90% confidence interval of each sweet.Galaxy Minstrels90% confidence interval = (sample mean – N.D.V * s.e, sample mean + N.D.V * s.e)= 2.61 – 1.645 * 0.017, 2.61 + 1.645 * 0.017= 2.58, 2.64This shows that there is a 90% confidence that the weight of a Galaxy minstrel weighs between 2.58g and 2.64g.Maltesers:90% confidence interval = (sample mean – N.D.V * s.e, sample mean + N.D.V * s.e)= 2.02 – 1.645 * 0.035, 2.02 + 1.645 * 0.035= 1.96g, 2.08gThis shows that there is a 90% confidence that the weight of a maltesers weighs between 1.96g and 2.08g.Now I will calculate the 95% confidence intervals.Galaxy Minstrels95% confidence interval = (sample mean – N.D.V * s.e, sample mean + N.D.V * s.e)= 2.61 – 1.96 * 0.017, 2.61 + 1.96 * 0.017= 2.58, 2.64This shows that there is a 95% confidence that the weight of a Galaxy minstrel weighs between 2.58g and 2.64g.Maltesers:90% confidence interval = (sample mean – N.D.V * s.e, sample mean + N.D.V * s.e)= 2.02 – 1.96 * 0.035, 2.02 + 1.96 * 0.035= 1.95g, 2.09gThis shows that there is a 95% confidence that the weight of a maltesers weighs between 1.95g and 2.09g.Now I will calculated 99% confidence intervalsGalaxy Minstrels99% confidence interval = (sample mean – N.D.V * s.e, sample mean + N.D.V * s.e)= 2.61 – 2.575 * 0.017, 2.61 + 2.575 * 0.017= 2.57, 2.65This shows that there is a 99% confidence that the weight of a Galaxy minstrel weighs between 2.57g and 2.65g.*****Maltesers:70% confidence interval = (sample mean – N.D.V * s.e, sample mean + N.D.V * s.e)= 2.02 – 0.525 * 0.035, 2.02 + 0.525 * 0.035= 2.00g, 2.04gThis shows that there is a 70% confidence that the weight of a maltesers weighs between 2.00g and 2.04g.