# From Samples to Populations

FROM SAMPLES TO POPULATIONS 5

FromSamples to Populations

Unit8: – Discussion

Part1: Confidence IntervalsWhyis it often impossible to know the actual value of any populationparameter?

Thesample estimates of the population parameters differ from the truepopulation values because these estimates are subject to samplingerrors. These are errors that arise due to the sampling procedure. These errors cannot be detected and thus cannot be eliminated fromany research work.

Givean example of a population parameter that you cannot calculate, butthat you can estimate.

Theproportion of defective items in a production process for the checkupof quality acceptance in the final goods produced (McCallum, 2007).Asample can be used to estimate a population parameter. How does thesample size affect the estimate?

Thesample size is a very significant feature of any research study inwhich the objective is to make conclusions about apopulation ofinterest from a sample. Therefore, the size of the sample needs tohave a satisfactory significance power.Large sample sizes generallylead to improvedprecision of the estimates of the unknown parameters.

Ifthe sample is larger, what will this do to the error E?

Fromthe formula above, larger sample sizes leads to a reduction in theerror of any empirical study.Usethe Confidence Interval formula above and calculate the 95%confidence interval for any population mean of your choice. Writedown (invent) the sample size (be sure it is 30 or above), the samplemean, and the sample standard deviation. Then, calculate theconfidence interval. Remember, you are inventing all the values, sono two posts should look the same.Use Excel and your inventedvalues to calculate the confidence interval. Include and compare theresults. (Tutorials can be found in Doc Sharing). Again, rememberthat your sample size must be 30 or above.

Theinvented values are:

Samplemean,and

The95% confidence interval for the population mean using the abovesample values is given by

Weare 95% confidence that the true population mean lies between113.399667 and 116.6003333.

Question23: Time to Graduation. Data from the national center for educationstatistics on 4,400 college graduates show that the mean timerequires to graduate with a bachelor’s degree is 5.15 years withstandard deviation of 1.68 years. Use a single value to estimate themean time required to graduate for all college graduates. Also findthe 95% confidence interval.

• The sample mean is the estimate of the population mean

Therefore,the single value estimate of themean time required to graduate for all college graduates

• The 95% confidence interval is given as

Samplemean – E &ltpopulation mean &lt Sample mean + E

Sincethe population variance is unknown, we use the t-distribution. Here,

Thus,the 95% confidence interval is

Weare sure that the true population mean falls between 5.100346 and5.199654.

References

McCallum,H. (2007). PopulationParameters: Estimation for Ecological Models.Chichester: John Wiley &amp Sons.