In class we conducted a sampling exercise.The population consisted of 25 elements.The possible values for the elements in the population ranged from 1 to 9 and were distributed as follows:ValueFrequencyProportionPercentage110.044%220.088%330.1212%440.1616%550.220%640.1616%730.1212%820.088%910.044%As in the class exercise, 10 elements are drawn replacing each element before the next is drawn.This was done 1000 times.The data set is given in a separate spreadsheet and consists of the numbers drawn for these 1000 draws in the order in which these were drawn. Each line can be considered aWith all calculations, include the formula that you used. Let’s start with a few questions about the population. The population is the complete set of elements from which the samples are drawn.What is the size of the population?The size of the population is 9Is the population finite or infinite?The population is finite since 9 is finite.Is the population discrete or continuous?The population is discrete.What is the level of measurement? Is it nominal, ordinal, interval, or ratio?The level of measurement used is ratioWhat is the mean of the population?What is the median of the population?What is the standard deviation of the population?Give an appropriate graphical representation of the population.What can be said of the shape of the population? Sampling from the population.What sampling strategy was used?Were the samples taken with or without replacement?The samplings were taken with replacement.What is the size of the samples?Are the samples considered to be big or small? What are the implications of this? Using Excel, compute the mean and standard deviation for each of the samples. Each sample will have a mean and a standard deviation.What, theoretically, would be the highest sample mean that could ever be observed?What was the highest sample mean actually observed?What, theoretically, would be the smallest sample mean that could ever be observed?What was the smallest sample mean actually observed?What, theoretically, would be the highest sample standard deviation that could ever be observed?What was the highest sample standard deviation actually observed?What, theoretically, would be the smallest sample standard deviation that could ever be observed?What was the smallest sample standard deviation actually observed? The distribution of sample means.How many samples were drawn from this population? What is this distribution of sample means called?Is it a theoretical or empirical sampling distribution?Is this distribution of sample means continuous or discrete?The ‘event’ is that the mean of a sample is equal to a given value. There are many samples that will have the same mean. How many different ‘events’ are possible?What is the mean of the observed distribution of sample means?What is the standard deviation of the observed distribution of sample means?How does the mean of the distribution of sample means compare to the population mean?How does the standard deviation of the distribution of sample means compare to the population standard deviation? These next questions require drawing a bar chart for the sample means. Please include this bar chart as part of your assignment.Create an appropriate graph of the distribution of sample means. How does the mean of the distribution of sample means compare to the mean of the population?How does the standard deviation of the distribution of sample means compare to the standard deviation of the population?How does the shape of the distribution of sample means compare to the shape of the population?Construct a box plot for the observed distribution of sample means.What is the probability that a sample mean will be greater than 6? Let’s now use formulas to calculate the mean and standard deviation of the sampling distribution for the mean.Using the formula, what would be the expected or theoretical value for the mean of the sampling distribution of the mean?Using the formula, what would be the expected or theoretical value for the standard deviation of the sampling distribution of the mean?How do these theoretical values for the mean and standard deviation of the sampling distribution of the mean compare to the empirical values for the mean and standard deviation of the sampling distribution of the mean we calculated earlier? More samples.What difference would taking more samples (say 5000 instead of 1000) have on the mean, standard deviation, and shape of the distribution of sample means?As n increases, standard deviation decreases. This is because the variance of the population increases. The mean is closer to the true value of the population. Bonus questionIf sampling had been done without replacement, what impact would this have had on the mean and the standard deviation of the distribution of sample means?It would make the standard deviation even lower than it would be with replacement. Without replacement makes it decrease even further.

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MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Question

In class we conducted a sampling exercise.

The population consisted of 25 elements.

The possible values for the elements in the population ranged from 1 to 9 and were distributed as follows:

Value	Frequency	Proportion	Percentage
1	1	0.04	4%
2	2	0.08	8%
3	3	0.12	12%
4	4	0.16	16%
5	5	0.2	20%
6	4	0.16	16%
7	3	0.12	12%
8	2	0.08	8%
9	1	0.04	4%

As in the class exercise, 10 elements are drawn replacing each element before the next is drawn.

This was done 1000 times.

The data set is given in a separate spreadsheet and consists of the numbers drawn for these 1000 draws in the order in which these were drawn. Each line can be considered a

With all calculations, include the formula that you used.

Let’s start with a few questions about the population. The population is the complete set of elements from which the samples are drawn.
What is the size of the population?

The size of the population is 9

Is the population finite or infinite?

The population is finite since 9 is finite.

Is the population discrete or continuous?

The population is discrete.

What is the level of measurement? Is it nominal, ordinal, interval, or ratio?

The level of measurement used is ratio

What is the mean of the population?
What is the median of the population?
What is the standard deviation of the population?
Give an appropriate graphical representation of the population.
What can be said of the shape of the population?
Sampling from the population.
What sampling strategy was used?
Were the samples taken with or without replacement?

The samplings were taken with replacement.

What is the size of the samples?
Are the samples considered to be big or small? What are the implications of this?
Using Excel, compute the mean and standard deviation for each of the samples. Each sample will have a mean and a standard deviation.
What, theoretically, would be the highest sample mean that could ever be observed?
What was the highest sample mean actually observed?
What, theoretically, would be the smallest sample mean that could ever be observed?
What was the smallest sample mean actually observed?
What, theoretically, would be the highest sample standard deviation that could ever be observed?
What was the highest sample standard deviation actually observed?
What, theoretically, would be the smallest sample standard deviation that could ever be observed?
What was the smallest sample standard deviation actually observed?
The distribution of sample means.
How many samples were drawn from this population?

What is this distribution of sample means called?
Is it a theoretical or empirical sampling distribution?
Is this distribution of sample means continuous or discrete?
The ‘event’ is that the mean of a sample is equal to a given value. There are many samples that will have the same mean. How many different ‘events’ are possible?
What is the mean of the observed distribution of sample means?
What is the standard deviation of the observed distribution of sample means?
How does the mean of the distribution of sample means compare to the population mean?
How does the standard deviation of the distribution of sample means compare to the population standard deviation?
These next questions require drawing a bar chart for the sample means. Please include this bar chart as part of your assignment.
Create an appropriate graph of the distribution of sample means.

How does the mean of the distribution of sample means compare to the mean of the population?
How does the standard deviation of the distribution of sample means compare to the standard deviation of the population?
How does the shape of the distribution of sample means compare to the shape of the population?
Construct a box plot for the observed distribution of sample means.
What is the probability that a sample mean will be greater than 6?
Let’s now use formulas to calculate the mean and standard deviation of the sampling distribution for the mean.
Using the formula, what would be the expected or theoretical value for the mean of the sampling distribution of the mean?
Using the formula, what would be the expected or theoretical value for the standard deviation of the sampling distribution of the mean?
How do these theoretical values for the mean and standard deviation of the sampling distribution of the mean compare to the empirical values for the mean and standard deviation of the sampling distribution of the mean we calculated earlier?
More samples.
What difference would taking more samples (say 5000 instead of 1000) have on the mean, standard deviation, and shape of the distribution of sample means?

As n increases, standard deviation decreases. This is because the variance of the population increases. The mean is closer to the true value of the population.

Bonus question
If sampling had been done without replacement, what impact would this have had on the mean and the standard deviation of the distribution of sample means?

It would make the standard deviation even lower than it would be with replacement. Without replacement makes it decrease even further.

Definition Definition Middle value of a data set. The median divides a data set into two halves, and it also called the 50th percentile. The median is much less affected by outliers and skewed data than the mean. If the number of elements in a dataset is odd, then the middlemost element of the data arranged in ascending or descending order is the median. If the number of elements in the dataset is even, the average of the two central elements of the arranged data is the median of the set. For example, if a dataset has five items—12, 13, 21, 27, 31—the median is the third item in ascending order, or 21. If a dataset has six items—12, 13, 21, 27, 31, 33—the median is the average of the third (21) and fourth (27) items. It is calculated as follows: (21 + 27) / 2 = 24.

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