Sampling Distribution of Sample proportion
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This document discusses sampling distributions of sample proportions. It provides examples of how to calculate the probability that a sample proportion will fall within a certain range of the true population proportion. In one example, a candidate claims 53% of students support her candidacy. The document calculates the probability that a random sample of 400 students will show less than 49% support as 5.48%. Another example calculates the probability that a candidate with actual 80% support will receive over 50% in a sample of 100 students as 77.34%, indicating a good chance of meeting the majority requirement.
Sampling Distribution of Sample proportion
- 1. Sampling Distribution of Sample Proportion Unit 5 This is the last part of Unit 5
- 3. Sampling Distribution of Sample Proportion Assume that in an election race between Candidate A and Candidate B, 0.60 of the voters prefer Candidate A. If a random sample of 10 voters were polled, it is unlikely that exactly 60% of them, or 6, would prefer Candidate A. By chance the proportion in the sample preferring Candidate A could easily be a little lower than 0.60 or a little higher than 0.60.
- 4. Sampling Distribution of Sample Proportion The sampling distribution of p is the distribution that would result if you repeatedly sampled 10 voters and determined the proportion p that favored Candidate A. Of course it did not have to be 10! 10 was only the sample size which we can call n and N is the total number in the population.
- 5. Central Limit Theorem According to the Central Limit Theorem, the sampling distribution of p-hat is approximately normal for sufficiently large sample size, such that np > 5 and nq > 5.
- 6. Calculations The formulae for calculating these proportions are given as follows: p = X/N and Ƹ 𝑝 = x/n where: ◦ X is the number of elements in the population possessing the characteristic of interest; N is the total number of elements in the population; ◦ x is the number of elements in the sample possessing the characteristic of interest; and n is the total number of elements in the sample. The distribution of p is closely related to the Binomial distribution.
- 7. Mean and Standard Deviation of p
- 8. Let us look at some examples… On page 75 of your unit notes, there is Example 5.1: We have five students A, B, C, D and E and we are interested in knowing which of these took statistics in sixth form. It turns out that A , D and E did. The others not. Based on this information, we have the proportion of students who took statistics in sixth form is P = X/N = 3/5 = 0.60 If we select all possible samples of the students who took statistics in sixth form we have:
- 9. Statistics in Sixth Form
- 10. Example 5.3: Ms. Smarthead Imagine that Ms. Smarthead is running for president of the Guild of Students at the St. Augustine campus of UWI. She claims that 53% of students favour her as their choice for the post of president. Assume that this claim is true. What is the probability that in a random sample of 400 students at the St. Augustine campus, less than 49% will favour her as their choice for president?
- 11. Imagine that Ms. Smarthead is running for president of the Guild of Students at the St. Augustine campus of UWI. She claims that 53% of students favour her as their choice for the post of president. Assume that this claim is true. What is the probability that in a random sample of 400 students at the St. Augustine campus, less than 49% will favour her as their choice for president? From the information given, we know that the population proportion p = 0.53 (53%) and therefore q = 0.47. We shall use ෝ 𝑝 = 0.49
- 12. Imagine that Ms. Smarthead is running for president of the Guild of Students at the St. Augustine campus of UWI. She claims that 53% of students favour her as their choice for the post of president. Assume that this claim is true. What is the probability that in a random sample of 400 students at the St. Augustine campus, less than 49% will favour her as their choice for president? We are looking for the probability that less than 49% will favour her as their choice of president. The probability we are interested in is finding P( Ƹ 𝑝 < 0.49). Therefore, the corresponding z value is z = (0.49 – 0.53)/0.025 = -1.60. Hence, P( ෝ 𝑝 < 0.49) = P( 𝑧 < -1.60) = 0.0548. So there is only a small probability that less than 49% will favour her as president. We can conclude that the probability that in a random sample of 400 students at the St. Augustine campus, less than 49% will favour her as their choice for president is 0.0548. So, she has a pretty good chance of being elected president.
- 13. Looking at the problem another way… Imagine that Ms. Smarthead is running for president of the Guild of Students at the St. Augustine campus of UWI. She claims that 80% of students favour her as their choice for the post of president. Assume that this claim is true. What is the probability that in a random sample of 100 students at the St. Augustine campus, she will be selected if the requirement is at least a simple majority in favour will confirm her as their choice for president?
- 14. Solution This is another example of Sampling Distribution of Sample Proportion. 80% of students favour her as their choice for the post of president so p = 0.80 and q = 0.20 with n = 100. Std dev, σ Ƹ 𝑝 = √(0.80 * 0.20)/100 = 0.04 The z-value for (0.50 - 0.80)/0.04 = -0.75 We are looking for P( Ƹ 𝑝 > 0.50) = P(z > -0.75) = 77.34% so she has a good chance.
- 17. Questions