# 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

Z=

2006) =0.8849Probability (mean

2006) =1-0.8849=0.1151 Probability (mean

For sample n=40

Mean=1993.88

Std dev.=7.781

Z=

2006) =0.9394Probability (mean

2006) =1-0.9394=0.0606 Probability (mean

For sample n=80

Mean=1996.37

Std dev.=11.408

Z=

2006) =0.799Probability (mean

2006) =1-0.799=0.201 Probability (mean

For sample n=160

Mean=1996.89

Std dev.=9.527

Z=

2006) =0.8315Probability (mean

2006) =1-0.8315=0.1685 Probability (mean

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

Sample n=10

Z=

2002) =0.7823Probability (mean

Z=

1990) =0.3121Probability (mean

2002) =0.7823-0.3121= 0.5109 Probability (1990<mean

Sample n=40

Z=

2002) =0.8508Probability (mean

Z=

1990) =0.3121Probability (mean

2002) =0.8508-0.3121= 0.5387 Probability (1990<mean

Sample n=80

Z=

2002) =0.6879Probability (mean

Z=

1990) =0.2877Probability (mean

2002) =0.6879-0.2877= 0.4002 Probability (1990<mean

Sample n=160

Z=

2002) =0.7054Probability (mean

Z=

1990) =0.3050Probability (mean

2002) =0.6879-0.3050= 0.3829 Probability (1990<mean

* 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.

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

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.

Figure 23