Distribution of the weights of two types of sweets

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
• Maltesers

When 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, usually

done 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.

2.05

2.51

2.45

1.76

1.60

1.60

2.24

1.73

2.43

2.65

1.70

2.10

1.89

2.18

1.62

1.44

2.03

2.18

2.20

1.88

1.64

1.95

2.25

1.87

1.71

2.11

1.95

2.27

2.08

2.38

1.87

2.27

2.02

2.15

1.43

1.69

2.22

1.70

2.46

1.48

2.15

2.35

1.89

1.69

1.16

2.19

2.04

2.66

2.01

2.33

2.09

2.31

2.27

1.86

2.23

1.89

1.97

2.76

2.04

2.14

2.33

2.15

1.73

1.34

1.67

1.74

1.92

1.76

2.41

2.13

1.49

1.80

1.96

1.91

2.55

2.49

2.51

1.85

2.06

1.40

2.02

2.07

2.09

2.16

2.18

Mean, Variance and Standard deviation

Now 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 = ∑x

      n

I will calculate the mean of the Galaxy minstrel’s and Maltesers separately.

Galaxy Minstrels u =    ∑x    =   261.41  = 2.61g (2 d.p)

                              n                 100

Maltesers, u =   ∑x   =   202.32   = 2.02g (2 d.p)

                      n                 100

Variance

The variance, o  ², is calculated by taking the mean of the sum of the squared deviations from the mean. The formula to calculate variance is:

o  ² = ∑x ²  - x ²

  n

I will once again calculate the variance of the Galaxy minstrels and Maltesers separately.

Galaxy Minstrels, o  ² = ∑x ²  - x ²  =  684.2055  - 2.61²

        n                  100

                                         = 6.842 – 6.812

 =0.03 (2 d.p)

Maltesers, o  ² = ∑x ²  - x ²  =  419.735  - 2.02²

       n                            100

                = 4.197 – 4.080

= 0.12 (2 d.p)

Standard Deviation

The standard deviation, o  , is the measure of spread around the mean. It is calculated using this formula:

o = /∑x ²  - x ²

  n

As you can see the standard deviation is simply the squared root of the variance:

√variance

As I have already calculated the variance it will be relatively easy to calculate the standard deviation (by square rooting the variance)

Galaxy Minstrels o  = √variance = √0.03

                                        = 0.17 (2 d.p)

Maltesers o  = √variance = √0.12

                                = 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:

                               = 2.61 – 1.645 * 0.017, 2.61 + 1.645 * 0.017

                               = 2.58, 2.64

This 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, 2.08g

This 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 Minstrels

95% 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, 2.64

This 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, 2.09g

This 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 intervals

Galaxy Minstrels

99% 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, 2.65

This 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, 2.04g

This shows that there is a 70% confidence that the weight of a maltesers weighs between 2.00g and 2.04g.

Table