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Let Us Study Dealing with various problems or situations oftentimes leads to confusion. A parameter to be tested must be identified carefully in order to formulate the hypotheses for population proportion. In this section, take note that problems involving proportions, unlike in population mean and sample mean, never use terms such as "average" and "mean" but "percentage" instead. Let us first differentiate population proportion and sample proportion. Population Proportion and Sample Proportion Definition Formula Population is a part of the proportion(p) population with a number of members in the particular attribute or population with a particular trait expressed as a attribute fraction, decimal, or P = number of members in percentage of the whole the population population. p x N ;P= % Sample is the ratio of the proportion (p) number of elements in the sample possessing number of members in the p - read as the characteristics of sample possessing the 'p hat" interest over the p = characteristics of interest number of elements in number of members in the sample the sample or n. p 18 n Consider the situation: In Matapat City, 10% of the residents are senior citizen. A survey was conducted to 500 randomly selected senior citizen residents to determine if they have cell phones. Out of 500 respondents, 421 answered that they own a cell phone.Notice that in Matapat City, 10% (percentage is used) of the entire residents are senior citizen. Therefore, the percentage of the senior citizen residents represents the population proportion or percentage which makes p = 10% = 0.10. Similarly, among these senior citizens, what percentage owns a cell phone? That illustrates the sample proportion, in symbol 15 (read as \"p hat\") which is computed as follows: no.0f senior citizen residents with cell phone I? no.0f senior citizen residents A _ 21 p 500 = 0.84 Sometimes, the sample proportion (15) is stated directly, such as: o \"20% of the respondents\" = 0.20 o \"5% of the defective bulbs\" = 0.05 o \"50% of the Grade 12 students\" = 0.50 On the other hand, there are cases where we still need to calculate 13. Examples of these kinds are: c \"70 out of 200 residents are married.\" 0 \"150 out of 500 listeners are interviewed.\" o \"10 out of 1000 bulbs are defective.\" In this case, we need to solve for the value of the sample proportion 13 (read as \"p hat\") using the formula, x 19:; where: 13 is the proportion of the number of successes in 71 samples and read as \"p hat\". x represents the number of \"successes\" in n samples; and n represents the size of the sample. Illustrative Example: For a class project, a Grade 12 STEM student wants to estimate the percentage of students in his school who are registered voters. From 45% Grade 12 students, he surveys 500 students and nds that 200 are registered voters. Determine the value of p and compute for the sample proportion. Solution: In words In symbol Population the rate or percent used from the p = 45% = 0. 45 Proportion entire Grade 12 students Sample number of registered Grade 12 students A _ 200 Proportion numer of surveyed Grade 12 students 2' 500 p = 0.4- or 40% Using the Central Limit Theorem in Testing Population Proportion When testing situations involving proportion, a percentage, or a probability, the following assumptions must be considered: 1. The conditions for binomial experiment are met. That is, there is a xed number of independent trials with constant probabilities and each trial has two outcomes that we usually classify as: \"success\" (p) and 'ffailure\" (q). The sum ofp and qis 1. Hence, we can writep+ q= 1 or q= 1 p. 2. The conditions up 2 5 and nq 2 5 are both satised (However, the specific number varies from source to source, some authors use 1 0 instead of 5 depending on how good an approximation one wants.) up 2 5 and nq Z 5 served as the basis to determine whether the sample size from the population proportion is sufciently large or not. Remember that this time, the condition that sample be large is not It to be at \"least 30\" but it should satisfy the second assumption. For a large size of sample proportions, the Central Limit Theorem (CLT) can be used. Bear in mind that if the sample size is sufciently large, then the mean of the random sample from a population has a sampling distribution that is approximately normal, even when the original distribution is normally distributed and n 2 30. Now, let us check the assumptions from the previous situation: 1. The responses have only two outcomes: \"registered voter\" (success) or \"not registered voter\" (failure). Therefore, the rst assumption is met. 2. To be able to satisfy the second condition, determine the value of p and n p = 0.45 n = 500. Solve for q: q = 1 p q=1045 q = 0.55. Through substitution, it shows that the second assumption is also met, since: np Z 5 and nq Z 5 500 (0.45) 2 5 and 500 (0.55) 2 5 225 2 5 and 275 2 5 Since we have shown that np 2 5 and nq 2 5, all conditions are met where the sample size is truly large enough to use CLT. In this condition, the test statistic to be used is the ztest statistic for proportions denoted by Zcom or the computed z-value. The z-Test Statistic for Population Proportion Yle 0r With np 2 5 and nq 2 5 and with the standard deviation of sample proportion be JE: , substitute 13 for 92 , p for [if and JE for m n Recall the zscore formula to be Z 2 Therefore, the formula for the value of ztest statistic for population proportion would be: is the z-test statistic for proportion. is the sample proportion ( i ). is the hypothesized value of the population proportion. is the sample size or the number of observations in the sample. is equal to 1 p