Hypothesis Test for a Single Proportion

imagesA hypothesis test for a single proportion measures the proportion of a sample taken from the population for a given trait in order to infer information about the proportion of that trait in the whole population itself.  The hypothesis test for a single proportion is a test of two contradictory hypotheses about the proportion of the characteristic under question in the population.  It begins with a hypothesis called a “null hypothesis” stating either the conventional wisdom if there is conventional wisdom on the subject, or the boring assumption that the trait in question is not prevalent.  The other hypothesis is called the “alternative hypothesis.”  This is the hypothesis that is hoped to be accepted.  Therefore, the null hypothesis is simply set up as the straw man for the purpose of being able to reject it in order that the alternate can be accepted. The philosophical concept behind the test can be stated as, “If the sample proportion obtained by random sampling is not very likely if the null hypothesis is true, then the null hypothesis must not be very likely to be true.”  In other words, the true proportion probably lies somewhere closer to the sample proportion.  This justifies rejecting the null hypothesis and accepting the alternative hypothesis. The test is accomplished in five steps: 1) H_{0}: [p=.x] (The null hypothesis states that the proportion "p" is equal to a decimal fraction ".x" of the population. Moreover, it will state in contextually meaningful words what trait the proportion is representing as the truth.  Additionally, the null hypothesis must be stated in such a way that it can be contradicted.) H_{A}: [p\neq .x, p<.x, p>.x] (The alternate hypothesis is what we hope to say is true about the population.  It must always be in contradiction to the null hypothesis. This is why the symbols show that the proportion is either not equal to, is less than, or is greater than the proportion stated in the null hypothesis.  Like the null hypothesis, the alternate hypothesis is also stated in contextually meaningful words.) 2) After stating the null and alternate hypothesis, some conditions must be analyzed in order to ascertain the validity of the test. (a) We must know that the sample is representative so that there is no bias in the test. (b) The sample can only have two outcomes.  That is a success or a failure. (c) The sample size must be large enough so that pn>10 and (1-p)*n>10, where p is the proportion given in the null hypothesis and n is the sample size. (d) In order to ensure that some data in the sample does not affect the outcome of another data in the sample, the sample size must be less than 10% of the population it is drawn from.  If the sample is greater than 10% of the population, we must be able to argue that all of the data in the sample is independent, such that no data predicts other data. 3) We must calculate a test statistic in order to know whether the sample is likely given the assumption that the null hypothesis is true.  The test statistic is denoted as: Z = \frac{ \hat{p} - p }{ \sqrt{ pq / n} }.                     In this formula, the sample proportion is denoted by \hat{p} and the assumed actual proportion, or proportion given in the null hypothesis, is denoted by pZ is the number of standard deviations away from the null proportion that the sample proportion is located. 4) We must now calculate the likelihood that the sample data is at least Z standard deviations from the null proportion if the null proportion is true.  If the alternate hypothesis states that the population proportion is not equal to the assumed proportion, we must calculate the likelihood of the sample being at least as extreme as our sample both to the right and left of the assumed proportion.  However, if the alternate hypothesis states that the population proportion is less than the assumed proportion, or greater than the assumed proportion, we can test for the likelihood of a sample proportion at least as extreme as ours either to the right or left of the assumed proportion (depending on whether our alternate hypothesis says the true proportion is less or greater).  With this in mind, if the likelihood given the null is less than an arbitrary value (This is called the significant or critical value and usually equal to .05.), the sample proportion is considered statistically significant and we are able to reject the null hypothesis. 5)  We must conclude by stating that we reject the null (if the sample proportion is statistically significant), or fail to reject the null.  This must be followed by a contextually meaningful statement of whether or not there is sufficient evidence to accept the alternate hypothesis. This link gives an informative discussion about the procedure for two-sided hypothesis testing for a single proportion, as well as for a mean.  There is also additional information integrated into this topic dealing with confidence intervals.  Moreover, I found this link having a useful bit about some pitfalls involved with hypothesis testing under the subheading “Cautions.”  Finally, I found this site to walk through specific examples of the two sided and the one sided hypothesis test.


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