Sampling Distribution of a Sample Proportion Lecture 26 Section 8.4 Robb T. Koether Hampden-Sydney College Mon, Mar 1, 2010 Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 1 / 33
Outline 1 The Central Limit Theorem for Proportions 2 Applications 3 Assignment Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 2 / 33
Outline 1 The Central Limit Theorem for Proportions 2 Applications 3 Assignment Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 3 / 33
The Central Limit Theorem for Proportions Theorem (The Central Limit Theorem for Proportions) For any population, the sampling distribution of ˆp has the following mean and standard deviation: µˆp = p σˆp = p(1 p). n Furthermore, the sampling distribution of ˆp is approximately normal, provided n is large enough. Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 4 / 33
The Central Limit Theorem for Proportions The Sample Size n is large enough if np 5 and n(1 p) 5. Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 5 / 33
Outline 1 The Central Limit Theorem for Proportions 2 Applications 3 Assignment Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 6 / 33
Applications Suppose that 60% of all high-school students own a cell phone. If we survey 150 high-school students, how likely is it that we will find that at least 65% of them own a cell phone? Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 7 / 33
Applications If p = 0.60 and our sample size is n = 150, then ˆp is normal with mean µˆp = 0.60 and σˆp = (0.60)(0.40) 150 = 0.0016 = 0.04. Want to know the probability that ˆp 0.65. Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 8 / 33
Applications 8 6 4 2 Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 9 / 33
Applications 8 6 4 2 Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 10 / 33
Applications The probability that ˆp is greater than 0.65 is normalcdf(.65,e99,.60,.04) = 0.1056. Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 11 / 33
Applications What if our sample size were 600 instead of 150? Then ˆp is normal with mean µˆp = 0.60 and σˆp = (0.60)(0.40) 600 = 0.0004 = 0.02. Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 12 / 33
Applications 15 10 5 Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 13 / 33
Applications 15 10 5 Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 14 / 33
Applications The probability that ˆp is greater than 0.65 is normalcdf(.65,e99,.60,.02) = 0.0062. Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 15 / 33
Applications Suppose that the true proportion of Americans who strongly disapprove of President Obama s performance is 40%. Can we use a sample of 1500 Americans to disprove the hypothesis that the rate is 45%? Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 16 / 33
Applications Which is the correct question to ask? Given that we observed ˆp = 0.40 in a sample, what is the probability that p = 0.45? If p really is 0.45, what is the probability that we would observe ˆp = 0.40 (or more extreme)? Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 17 / 33
Applications Which is the correct question to ask? Given that we observed ˆp = 0.40 in a sample, what is the probability that p = 0.45? If p really is 0.45, what is the probability that we would observe ˆp = 0.40 (or more extreme)? Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 18 / 33
Hypothesis Testing Let us test the hypotheses H 0 : 45% of all Americans strongly disapprove of President Obama s performance. H 1 : Less than 45% of all Americans strongly disapprove of President Obama s performance. Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 19 / 33
Hypothesis Testing Using the Central Limit Theorem, the null hypothesis predicts that the distribution of ˆp is Normal, with Mean p, which is 0.45 (hypothetically). Standard deviation (0.45)(0.55) 1500 = 0.0128. That is, ˆp is N(0.45, 0.0128). Find the probability that ˆp 0.40. Design a decision rule so that α = 0.05. Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 20 / 33
Homework Problem Exercise 8.12, page 528. Suppose that 60% of all students at a large university access course information using the Internet. (a) Sketch a picture of the distribution for the possible sample proportions you could get based on a simple random sample of 100 students. Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 21 / 33
Homework Problem Exercise 8.12, page 528. Suppose that 60% of all students at a large university access course information using the Internet. (a) Sketch a picture of the distribution for the possible sample proportions you could get based on a simple random sample of 100 students. 8 6 4 2 5 Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 21 / 33
Homework Problem Exercise 8.12, page 528. (b) Sketch a picture of the distribution for the possible sample proportions you could get based on a simple random sample of 100 students. (c) Use the 68 95 99.7 rule for normal distributions to complete the following statements: (i) There is a 68% chance that the sample proportion is between and. (ii) There is a 95% chance that the sample proportion is between and. (iii) It is almost certain that the sample proportion is between and. Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 22 / 33
Homework Problem Exercise 8.12, page 528. (b) Sketch a picture of the distribution for the possible sample proportions you could get based on a simple random sample of 100 students. (c) Use the 68 95 99.7 rule for normal distributions to complete the following statements: (i) There is a 68% chance that the sample proportion is between 0.551 and 0.649. (ii) There is a 95% chance that the sample proportion is between and. (iii) It is almost certain that the sample proportion is between and. Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 23 / 33
Homework Problem Exercise 8.12, page 528. (b) Sketch a picture of the distribution for the possible sample proportions you could get based on a simple random sample of 100 students. (c) Use the 68 95 99.7 rule for normal distributions to complete the following statements: (i) There is a 68% chance that the sample proportion is between 0.551 and 0.649. (ii) There is a 95% chance that the sample proportion is between 0.502 and 0.698. (iii) It is almost certain that the sample proportion is between and. Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 24 / 33
Homework Problem Exercise 8.12, page 528. (b) Use the 68 95 99.7 rule for normal distributions to complete the following statements: (i) There is a 68% chance that the sample proportion is between 0.551 and 0.649. (ii) There is a 95% chance that the sample proportion is between 0.502 and 0.698. (iii) It is almost certain that the sample proportion is between 0.453 and 0.747. Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 25 / 33
Homework Problem Exercise 8.12, page 528. (c) Would it be likely to observe a sample proportion of 0.50, based on a simple random sample of size 100, if the population proportion were 0.60? Explain. Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 26 / 33
Homework Problem Exercise 8.12, page 528. (c) Would it be likely to observe a sample proportion of 0.50, based on a simple random sample of size 100, if the population proportion were 0.60? Explain. (The question should ask how likely it is to observe a sample proportion at least as low as 0.50.) Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 26 / 33
Homework Problem Exercise 8.12, page 528. (c) Would it be likely to observe a sample proportion of 0.50, based on a simple random sample of size 100, if the population proportion were 0.60? Explain. (The question should ask how likely it is to observe a sample proportion at least as low as 0.50.) No. A sample proportion of 0.50 is more than 2 standard deviations from the mean. Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 26 / 33
Homework Problem Exercise 8.12, page 528. (d) Sketch a picture of the distribution for the possible sample proportions you could get based on a simple random sample of 400 students. Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 27 / 33
Homework Problem Exercise 8.12, page 528. (d) Sketch a picture of the distribution for the possible sample proportions you could get based on a simple random sample of 400 students. 15 10 5 5 Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 27 / 33
Homework Problem Exercise 8.12, page 528. (i) How does this picture differ from the one in part (a)? Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 28 / 33
Homework Problem Exercise 8.12, page 528. (i) How does this picture differ from the one in part (a)? It is only half as wide. Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 28 / 33
Homework Problem Exercise 8.12, page 528. (i) How does this picture differ from the one in part (a)? It is only half as wide. How will the increased sample size affect the range of values you gave in (i) (iii) of part (b) Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 28 / 33
Homework Problem Exercise 8.12, page 528. (i) How does this picture differ from the one in part (a)? It is only half as wide. How will the increased sample size affect the range of values you gave in (i) (iii) of part (b) They will each be only half as wide. Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 28 / 33
Outline 1 The Central Limit Theorem for Proportions 2 Applications 3 Assignment Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 29 / 33
Assignment Homework ) Read Sections 8.1-8.2, pages 499-508. Exercises 7-14, page 526. Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 30 / 33
Assignment Homework Answers 8. (a) 0.0264. (b) Yes. The probability of that is 3.117 10 6. 10. (a) The sampling distribution for the sample proportion of men is normal with mean 0.49 and standard deviation 0.04999. (b) 0.0548. Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 31 / 33
Assignment Homework Answers 12. (a) Draw a normal curve with mean 0.60 and standard deviation 0.04899. (b) (i) 0.55101 and 0.64899. (ii) 0.50202 and 0.69798. (iii) 0.45303 and 0.74697. (c) No. According to part (b)(ii), a proportion as low as 0.50 has only about a 2 1 2 % chance of occurring. (d) Sketch a normal curve with mean 0.60 and standard deviation 0.02449. (i) It is narrower. (ii) They will each be half as wide as before. Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 32 / 33
Assignment Homework Answers 14. (a) It is normal with mean 0.50 and standard deviation 0.02887. (b) ˆp = 0.56. p-value of 0.56 is 0.0188. (c) Reject H 0. A majority of shoppers favor longer shopping hours. Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 33 / 33