Question

The average weight of 237 babies born at Swedish hospital last year was 7.04 pounds with a standard deviation of .42 pounds. Generate the population and then take two samples via the following steps. Compare their means and standard deviations.

Step 1: Open an excel worksheet and enter “population” in A1, “number” in A2, birth in B1 and “weights” in B2. Type 1 in A3 and ENTER.

Step 2: Make A3 your active cell. From the Ribbon select the following sequence: HOME, FILL, and SERIES. From the Series dialog box, select COLUMNS and LINEAR. In the Step value box type 1 and in the Stop value text box type 237. Then click OK. This enters the numbers 1-237 in A3:A239.

Step 3: Make B3 your active cell. Click on FORMULAS, MORE FUNCTIONS and STATISTICAL. Select NORM.INV. and OK. Then in the PROBABILITY box type: RAND(). In the MEAN box type: 7.04 and in the STANDARD DEVIATION box type: .42 and OK. Make B3 your active cell. Place your mouse arrow on the lower right corner of B3. Make sure you have the thick black plus sign. Click the left mouse button twice rapidly. This automatically fills the cells in B3:B239 with a population of babies from which we can draw samples.

Step 4: Highlight B3 to B239. Right click on the column and select COPY.   Then go to “sheet 2.” Place your cursor in A1 and right click and choose PASTE SPECIAL and indicate VALUES. Click OK. This fixes the population in place.

Step 5: On “sheet 2” enter “Sample 3” in E1 and “Sample 30” in G1. To take the samples, make E3 your active cell. Select: DATA and DATA ANALYSIS. Then select SAMPLING and click OK. For the INPUT RANGE enter A1:A237. Click the LABELS box. Select RANDOM and enter 3 for the NUMBER OF SAMPLES. Click on the OUTPUT RANGE and type E3:E5. Click OK. This creates a random sample of 3 from the population of 237. (If Data Analysis doesn’t appear, click on the File in the upper left corner: - Options -Add-ins -manage box (select Excel Add-ins -Go. Select Analysis Toolpak and click OK.)   

Step 6: Now calculate the mean and standard deviation of your population. Make C3 your active cell and type: =AVERAGE(A1:A237). Hit ENTER. Make C4 your active cell and type: =STDEV.P(A1:A237).   To calculate the mean and standard deviation of your sample of 3, do the same making E7 your active cell for the mean and E8 your active cell for the standard deviation (using =STDEV.S instead of STDEV.P, because S indicates sample and P indicates population).

Step 7: Take a sample of 30 per the instructions in Step 5 beginning with G3 as your active cell. Calculate the mean and standard deviation per Step 6 (using STDEV.S).

Please manually respond to the following questions:

1. What is the definition of “sampling error?”
2. What is the sampling error for the mean and standard deviation of each sample?
3. What pattern do you observe?

Answer 1

The sampling error is defined as the

where is the Z score based on the confidence interval

and is the population standard deviation

n is the size of the sample.

Now, the sampling error of the first sample:

Here, we have Z for 95% confidence is 1.96 and the population sd is 0.4351 and n=3. Substituting these, we get the sampling error for the first sample(sample of 3): is

For the second sample, the sampling error is

Here we summarize the sampling error and the standard deviation of each sample:

Sample	SD	Sampling
size		Error
3	0.4691	0.4294
30	0.3385	0.1557

  

From the above table, we observe that the standard deviation and sampling errors reduce with increase in sample size. The sampling error reduce drastically as the sample size increaes. Hence, I learnt from this exercise that as we increas the sample size, the standard deviation and sampling errors can be reduced.

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I had reproduced the EXCEL exercise as directed in the exercise only for teh samples.

		Sample 3		Sample 30
7.03001		6.512782		7.327948
0.435099		7.387928		6.451742
		6.657523		7.20253
				7.241047
		6.852745		6.818519
		0.469099		6.422378
				7.053079
				6.546224
				6.451742
				6.823666
				7.245196
				6.91968
				7.263859
				7.486416
				7.02046
				6.676493
				6.711239
				6.787412
				6.637326
				7.424628
				7.620815
				6.800679
				7.470351
				7.251886
				7.267184
				6.91968
				7.175399
				7.324391
				7.263916
				6.920242
				7.017538
				0.338505