Math215 Generating a Sampling Distribution of Sample Means | Complete Solution

Generating a Sampling Distribution of Sample Means
In this assignment we’ll illustrate the idea of a sampling distribution in the case of a very small sample from a very small population.
The population will be the ages of six people from your family or friends. The parameter of interest will be the mean age, µ, of this population. First, identify the six people in your family or friend groups and record their ages. Then follow the steps below to generate a sampling distribution and answer the questions.

    Record the data of your population in the table below. (i.e. the ages of the 6 people in your family or friends. Population size is N=6)
Individual    1    2    3    4    5    6
Age                        
 
    Find the mean age, µ, and the standard deviation, σ, of this population.
µ = ______________
σ = ______________

    Now, take 10 different random samples of size n= 3 from this population. You can use your calculator (See the instructions in Appendix A at the end of this assignment document) OR use Excel (See the YouTube link in Appendix B) to generate 10 random samples of size 3. Calculate the mean of each sample. List the ages in each sample and their mean in the table below:

    Sample Items    Sample means, x ̅
1.        x ̅1 =
2.        x ̅2 =
3.        x ̅3 =
4.        x ̅4 =
5.        x ̅5 =
6.        x ̅6 =
7.        x ̅7 =
8.        x ̅8 =
9.        x ̅9 =
10.        x ̅10 =

    Without actually calculating
    Predict what the mean of 10 sample means from your table above should be: ________
    Predict what the standard deviation of 10 sample means from your table should be: _______

    You have a distribution of x ̅ with 10 sample means. Construct a dot plot, a boxplot or a stem plot for this distribution and describe the shape of it. Include your graph here.

    Using your calculator or Excel, calculate the mean and standard deviation of the 10 sample means in #3 above.
Mean of sample means: _________
Standard deviation of sample means: _________
How are these related to your predictions in #4? __________________________________
How are they related to the population mean, µ, and population standard deviation that you have in #2 above? __________________________________________

    Now, take 10 more random samples of size n= 3 from your population of size 6. (There are a total of 20 different samples possible of size n=3 from a population of size N=6. You’ve got 10 of those samples in #3 above and you’ll get 10 more.) List all 20 samples and their means in the table below:

    Sample Items    Sample means, x ̅        Sample Items    Sample means, x ̅
1.        x ̅1 =    11.        x ̅11 =
2.        x ̅2 =    12.        x ̅12 =
3.        x ̅3 =    13.        x ̅13 =
4.        x ̅4 =    14.        x ̅14 =
5.        x ̅5 =    15.        x ̅15 =
6.        x ̅6 =    16.        x ̅16 =
7.        x ̅7 =    17.        x ̅17 =
8.        x ̅8 =    18.        x ̅18 =
9.        x ̅9 =    19.        x ̅19=
10.        x ̅10 =    20.        x ̅20 =

    Without actually calculating
    Predict what the mean of 20 sample means from your table above should be: ________
    Predict what the standard deviation of 20 sample means from your table should be: _______

    You have a distribution of x ̅ with 20 sample means. Construct a dot plot, a boxplot or a stem plot for this distribution and describe the shape of it. Include your graph here. Discuss how it is different from the one in #5?

    Using your calculator or Excel, calculate the mean and standard deviation of the 20 sample means in #7 above.
Mean of sample means, μ_x ̅  =_________
Standard deviation of sample means, σ_x ̅  =_________
How are these related to your predictions in #8? __________________________________
How are they related to the population mean, µ, and population standard deviation, σ, that you have in #2 above? __________________________________________          

     Explain how your findings above are related to the Central Limit Theorem. (Reference Sec 6.3 in your textbook)                         

APPENDIX A:
GENERATING SIMPLE RANDOM SAMPLES USING TI-84+
The TI-84 PLUS Calculator may be used to generate a simple random sample of values.  Here, we assume that each item in the population is assigned an integer between 1 and N.  We will use the calculator to randomly generate numbers between 1 and N and will use the corresponding population items to comprise the simple random sample.  There are two steps to this procedure.  The first is to assign a value to the seed.  The seed is a number that the calculator uses to randomly generate other values.  The second step is to run a command that randomly generates values.
We choose any value, positive or negative, and assign it as the seed.  Each different seed will produce a unique sequence of randomly generated values.  We use the number 5 as the seed.  Enter 5, press  , and then enter the following keystrokes:       .
 
Suppose that there are 18 items in the population.  The command randInt(1, 18, 5) will randomly generate 5 values between 1 and 18.  Press   and scroll right to the PRB menu and select the randInt command.  Enter 1, comma, 18, comma, 5 and press  .  
 
Example: Suppose we need to draw a sample of 4 ages of employees from the following list of 10.
We use 6 as the seed.
    23
    35
    39
    56
    46
    43
    38
    26
    62
    45
Solution: We enter 6, press  and then       .
 
Next, we press   and scroll right to the PRB menu and select the randInt command.  Enter
randInt(1,10, 4) and press  .  The simple random sample consists of the employees
with numbers 5,2,4,9. From our list, these correspond to the ages 46, 35, 56, 62.

Every time you want to generate a new sample, you need to seed a new number and repeat this process.

APPENDIX B:

To generate a random sample using Excel, please view this video (3 minutes):

https://www.youtube.com/watch?v=q8fU001P2lI&spfreload=1