Math Methods investigation of statistics

  • Category:
    Mathematics
  • Document type:
    Assignment
  • Level:
    High School
  • Page:
    1
  • Words:
    671

* If a sample of n coins is taken, what is the likelihood that the mean of that sample is greater than 2006? Investigate for different n.

For this investigation, you will be using a population of 5 cent pieces and a computer-based random sampler to generate samples, to calculate statistics based on the samples and to discover the characteristics of sampling distributions in order to solve the following problem:

The Problem

The production date of 5 cent pieces in circulation is approximately negatively skewed with a mean of 1996 and standard deviation of 10 years.

For sample n=10

Mean=1994.65

Std dev.=9.416

Math Methods investigation of statisticsZ=

2006) =0.8849Math Methods investigation of statistics 1Probability (mean

2006) =1-0.8849=0.1151Math Methods investigation of statistics 3 Probability (meanMath Methods investigation of statistics 2

For sample n=40

Mean=1993.88

Std dev.=7.781

Math Methods investigation of statistics 4Z=

2006) =0.9394Math Methods investigation of statistics 5Probability (mean

2006) =1-0.9394=0.0606Math Methods investigation of statistics 7 Probability (meanMath Methods investigation of statistics 6

For sample n=80

Mean=1996.37

Std dev.=11.408

Math Methods investigation of statistics 8Z=

2006) =0.799Math Methods investigation of statistics 9Probability (mean

2006) =1-0.799=0.201Math Methods investigation of statistics 11 Probability (meanMath Methods investigation of statistics 10

For sample n=160

Mean=1996.89

Std dev.=9.527

Math Methods investigation of statistics 12Z=

2006) =0.8315Math Methods investigation of statistics 13Probability (mean

2006) =1-0.8315=0.1685Math Methods investigation of statistics 15 Probability (meanMath Methods investigation of statistics 14

2 The probability that mean of sample lies between 1990 and 2002

Sample n=10

Math Methods investigation of statistics 16Z=

2002) =0.7823Math Methods investigation of statistics 17Probability (mean

Math Methods investigation of statistics 18Z=

1990) =0.3121Math Methods investigation of statistics 19Probability (mean

2002) =0.7823-0.3121= 0.5109Math Methods investigation of statistics 21 Probability (1990<meanMath Methods investigation of statistics 20

Sample n=40

Math Methods investigation of statistics 22Z=

2002) =0.8508Math Methods investigation of statistics 23Probability (mean

Math Methods investigation of statistics 24Z=

1990) =0.3121Math Methods investigation of statistics 25Probability (mean

2002) =0.8508-0.3121= 0.5387Math Methods investigation of statistics 27 Probability (1990<meanMath Methods investigation of statistics 26

Sample n=80

Math Methods investigation of statistics 28Z=

2002) =0.6879Math Methods investigation of statistics 29Probability (mean

Math Methods investigation of statistics 30Z=

1990) =0.2877Math Methods investigation of statistics 31Probability (mean

2002) =0.6879-0.2877= 0.4002Math Methods investigation of statistics 33 Probability (1990<meanMath Methods investigation of statistics 32

Sample n=160

Math Methods investigation of statistics 34Z=

2002) =0.7054Math Methods investigation of statistics 35Probability (mean

Math Methods investigation of statistics 36Z=

1990) =0.3050Math Methods investigation of statistics 37Probability (mean

2002) =0.6879-0.3050= 0.3829Math Methods investigation of statistics 39 Probability (1990<meanMath Methods investigation of statistics 38

* As sample size, n, increases, what effect does this have on the probability that the mean of the sample lies between 1990 and 2002?

The probability that the mean of the sample lies between 1990 and 2002 becomes small as the sample increases.

* Explain why the two situations have different outcomes.

Because the data is negatively skewed as the data is increased the mean moves to the right that reducing the probability.

Math Methods investigation of statistics 40

Figure 1: distribution of samples

* What outcomes do you think would be observed with any sampling distribution from any population with any mean and any standard deviation? Describe the conjecture.

In order to achieve these aims you will need to:

* Use a population of five cent pieces as a starting point. Describe the population in terms of shape and record the population mean and population standard deviation.

The population mean=50.08

Standard deviation =2.88

* Select ten random samples of 5 coins from the above population and record the sample mean and the sample standard deviation each time. (Use your Graphics Calculator).


* Create a dot plot of all the sample means gathered by the class. Describe the distribution and record the mean of all the sample means and the standard deviation of all the sample means.

Means of Ten random samples of 5 coins

S.D of ten random samples of 5 coins

Mean of sample means =2010.5

S.D of sample means=3.902

* Repeat the above 2 dot points but use ten samples of 10 coins.

Mean dot plot for 10 samples

S.D of ten random samples of 10 coins

10.06700

11.29848

Mean of sample means =2002.37

S.D of sample means=4.97

.Discuss your findings in terms of the problem above


Then:

* Use the DEMO on pg 65 of your Textbook to find the Population mean, Population standard deviation, Mean of the sample means and Standard deviation of the sample means for each sample size. Describe how the shape of the histograms change as the sample size increases.
It can be seen that the standard deviation is increasing towards the population SD of 11.7 when the coins are increased to 10 also the mean is tending towards the population mean as the samples are increased to 10 .

48.45889

Math Methods investigation of statistics 41
Figure 2 Histograms for different the sample sizes


* Using your graphics calculator, determine whether a linear, exponential or power model best shows the relationship between the sample size and the sample standard deviations for these sample sizes. Provide sufficient evidence for your choice.

The linear model shows the relationship as can be seen in figure 3. This is the best curve as it brings the all the points averagely close the curve.

Math Methods investigation of statistics 42 Figure 23