Sampling is thebest method of studying the population. It offers an easier way toanalyze different parameters of the population without having toconduct research on all the group. However, as a representation ofthe population, a sample has many shortcomings and may not necessarybe the accurate description of the group (Krishnamoorthy, 2016). Thepaper will discuss the margin of error and chi-square distributionusing working examples.
Question#1:Margin of Error of the Interval
Public opinionpolls fall within a certain margin of error. In other words, inpresenting the data, the pollster knows the statistic underconsideration fall between the sample mean and the true populationaverage. Another key factor in the presentation is the confidenceinterval, defines the uncertainty of the calculated value(Krishnamoorthy, 2016). For example, a company wants to decide on thebrand name to use for a new product. The first brand name is BigM andthe other FunA. The potential market for this new product is about400,000. A randomized sample is used to collect data on which brandname majority identified with. The results showed 60% of respondentsidentify with Fun A as compared to BigM (40%). The confidence levelis 95%. The calculated margin of error is 4%. The result means, theresearcher is 95% sure, 56 to 60% of the potential customers willidentify with FunA.
The primary use of chi-square is in probability, and the method hassignificant application in genetics. For example, it can show theextent to which a certain disorder or quality is inherited in thepopulation. However, the chi-square has found its use in all thefields of studies.
Normally, thedifference in the association is referred to as statisticallysignificant or insignificant. If the difference is negligible, thecalculated value falls within the range of the expected values at acertain level of confidence say 95%. (Krishnamoorthy, 2016). On theother hand, if the calculated value lies outside the expected value,then the difference is referred to as statistically significantshowing no association between the observed and expected value.
For example, 256fans were surveyed for their possibility to cheer their teams wildly.The results of the study were Arsenal (29), Chelsea (24), Wolves(22), Bolton (19), Leister (20), Swansea (23), Blackpool (18),Barcelona (20), and Juventus (23). The hypothesis to test is thatwild cheering is evenly distributed across all the clubs. If thedistribution was to be even for the sample, then the average shouldbe 21.33. The mean is what one would expect in the case of evendistribution (Krishnamoorthy, 2016). The next step involves findingthe difference between the expected and observed value. For example,in the case of Arsenal Fc the difference is 7.667 and for Bolton-2.33. Since the differences are both positive and negative, they areconverted to an absolute value by squaring. The absolute value of-2.33 is 5.44. In the next step, the absolute value is divided by theexpected value, in the case of Bolton, the figure is 0.255. All thevalues from the last step mentioned are summed up to give thechi-square value. In this case, it is 5.09. Assuming the degree offreedom in the chi-square for the question is 11 (Krishnamoorthy,2016). The p-value (confidence level) defines the range within whichthe calculated value lies. In this case, the value is 93.65%.
In conclusion, asmall p-value usually less than 5%, implying the confidence level is95% means the difference is statistically insignificant. On the otherhand, if the value falls outside 95%, it is considered statisticallysignificant. The purpose of the margin of error is to define theborders within which the calculated statistic lies.
Krishnamoorthy,K. (2016). Handbook of statistical distributions withapplications. CRC Press.
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