OBJECTIVES:

• To experimentally generate a sampling distribution of the sample mean.
• To determine how sample size impacts the sampling variability.
• To empirically verify the Central Limit Theorem in a real-life situation.

EXCEL PROCEDURES:

A company produces bags of packing peanuts having counts that are normally distributed with a mean 100 pieces and standard deviation of 15 pieces. We want to take 100 samples from this population of size n =5, and investigate the distribution of the sample means ().

1. In cell A1 type =ROUND(NORMINV(RAND( ), 100, 15),0). This formula is generating a random number (rounded to the nearest whole) from a normally distributed variable with a population mean of 100 and a population standard deviation of 15.

1. Click on cell A1. Hover the cursor over the small square in the lower right corner of cell A1 until the cursor turns to a black + sign. Left click and drag across to cell E1. You have just randomly generated a random sample of 5 IQ's from the distribution of IQ's.

1. To generate 100 random samples of size n=5, make sure that cells A1 through E1 are selected, then hover the cursor over the small square in the lower right corner of cell E1 until the cursor turns to a black + sign. Left click and drag down to cell E100. You now have 100 samples of size n=5 from the distribution of IQ's.

1. We are interested in the mean of each sample. To calculate the mean of the first sample, click on cell F1 and type =average(A1:E1). Hover the cursor over the small square in the lower right corner of cell F1 00 for later when we draw a histogram.

1. Click on the F at the top of Column F so that the entire column is highlighted. Right click and select Copy. Click on the G at the top of Column G so that the entire column G is highlighted. Right click and select Paste Special. Select Values and then click on OK.

1. Click on the G at the top of Column G so that the entire column is highlighted. At the top right choose "Sort" and select Smallest to Largest. When the prompt asks, choose continue with the current selection. Click OK.

1. Click on cell H1. Type =ROUND(G1,0). This will round G1 to the nearest whole. Autofill down using the hover + described above so that cells H2 through H100 are filled. You should now have 100 samples means, rounded to the nearest whole.

1. Next, we want to find the mean and standard deviation of the sample means.

1. Click on cell I12 and type =average(G1:G100).This is calculating the mean of the sample means. Record this value rounded to three places99.278

1. Click on cell I13 and type =stdev(G1:G100).This is calculating the standard deviation of the sample means (standard error). Record this value rounded to three places 6.035Next, we want to examine the distribution of sample means by constructing a histogram.

1. First we will fill in the following frequency table by hand. To fill in the frequency column, count the number of observations in each class from column H in your spreadsheet. Check to be sure your frequency column total matches the sample size.

Class	Midpoint	Frequency
73-78	76	0
79-84	82	0
85-90	88	6
91-96	94	27
97-102	100	34
103-108	106	26
109-114	112	6
115-120	118	1
121-126	124	0

1. In cell J1 type Midpoints.
2. In cells J2-J10 enter the midpoints from your frequency table above.
3. In cell K1 type Frequency.
4. In cells K2-K10 enter the frequencies for each class from your frequency table above.
5. Select cells K2-K10 by clicking and holding the left mouse button in cell K2 and dragging down to K10.
6. Select the Insert tab at the top of Excel.
7. Select the Column icon in the Charts group.
8. Select the left most icon under 2-D Column
9. Right click in the white area around the chart of the graph the pops up.
10. Click on Select Data from the menu that appears.
11. Select Edit in the box with the heading Horizontal Axis Labels.
12. The box that appears will prompt you to select a data range. Do this by highlighting cells J2-J10. Click OK, OK to close the boxes.
13. Click in the center of your graph to select it and select Quick Layoutsin the Design Tab.
14. Select Layout 8 (3^rd row, 2^nd column)
15. Click on the text Axis Title on both the x and y axis and rename these appropriately. (x: sample mean, y: frequency)
16. Right click on the bars in your chart and select Format Data Series from the popup menu.
17. Select Fill from the list on the left and check the box next to Vary Colors By Point
18. Select Border Color from the list on the left and check Solid Line. Click Close.
19. Click on the graph title and change to 'Sampling Distribution' and add your name.

please i need help in the analysis part only .

ANALYSIS:

To investigate the impact of sample size on sampling variability, generate 100 samples of size 10 in columns L through U and calculate the sample means in column V using the procedure as described above. We won't worry about fixing and rounding the values this time since we are not drawing a histogram.

In cell W12, use =average (V1:V100) to calculate the mean of the sample means.

Record this value ___

In cell W13, use =stdev (V1:V100) to calculate the standard deviation of the sample means (standard error). Record this value _______

1. What is the shape of the distribution of the sample meansfrom your histogram?

1. Based on the Central Limit Theorem, how do you expect the mean of the sample means to compare to the population mean? __________________________________

1. What is the value of the population mean that was stated at the beginning of the lab? ___________

1. What is the experimental value of the mean of the sample means (size n= 5) from cell I12? _____

1. What is the experimental value of the mean of the sample means (size n= 10) from cell W12? ___

1. Do you expect the standard deviation of the sample means (standard error) to increase or decrease as the sample size increases? __________

1. What is the experimental value of the standard deviation of the sample means (standard error) from cell I13? _______________

1. What is the experimental value of the standard deviation of the sample means (standard error) from cell W13? _____________

1. What is the value of the population standard deviation that was stated at the beginning of the lab? ___________
   1. What is the theoretical value of the standard error () when n = 5?___________
   2. What is the theoretical value of the standard error when n= 10?___________

1. How do these values compare to the experimental values in I13 and W13? Explain.

1. State the Central Limit Theorem. Do your answers to question #1, #2, #3, and #4 validate the Central Limit Theorem?

1. In the original problem we assumed the count in the packages were normally distributed. If the original distribution was not normal, how large of a sample is needed to apply the Central Limit Theorem?