# VCU STAT 210 - Lecture27 (73 pages)

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## Lecture27

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- Pages:
- 73
- School:
- Virginia Commonwealth University
- Course:
- Stat 210 - Basic Practice of Statistics

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STAT 210 Lecture 27 Confidence Intervals for Population Proportions October 30 2017 Test 5 Monday November 6 Covers the chapter on proportions and relevant concepts from chapter VII Combination of multiple choice fill in the blank questions and problems written questions Formulas and tables provided please bring calculator and writing instrument Practice Problems Sailboat Pages 252 through 257 Relevant problems IX 8 IX 12 a and b IX 13 IX 15 a and b IX 16 IX 17 and IX 18 Recommended problems IX 13 and IX 18 Hummingbird Pages 210 through 215 Relevant problems VIII 8 VIII 12 a and b VIII 13 VIII 15 a and b VIII 16 VIII 17 and VIII 18 Recommended problems VIII 13 and VIII 18 Additional Reading and Examples Sailboat Read pages 248 through 251 Pay particular attention to pages 249 and 250 Hummingbird Read pages 206 through 209 Pay particular attention to pages 207 and 208 Top Hat Statistical Inference Statistical inference involves using statistics computed from a sample data to make statements about unknown population parameters Statistical Inference Statistical inference involves using statistics computed from a sample data to make statements about unknown population parameters This chapter we want to make inferences about the population proportion p Example 1 Of interest is to estimate the proportion of all students at this university who have children What is the population of interest What is the parameter of interest Example 1 Of interest is to estimate the proportion of all students at this university who have children In this situation the population of interest is all students at this university and the parameter of interest is p the proportion of all students at this university with children Example 1 Of interest is to estimate the proportion of all students at this university who have children In this situation the population of interest is all students at this university and the parameter of interest is p the proportion of all students at this university with children It likely is not possible to determine the exact value of p and hence it will need to be estimated Point Estimate The point estimate of the population proportion p is the sample proportion p p number of successes in the sample sample size n Confidence Interval for p Goal Estimate the unknown population proportion p Point Estimate Sample proportion p Situation While p should be close to p it is very unlikely that p will equal p exactly Confidence Interval for p Goal Estimate the unknown population proportion p Point Estimate Sample proportion p Situation While p should be close to p it is very unlikely that p will equal p exactly Therefore to the point estimate we subtract and add a margin of error to create a 100 C confidence interval estimate for p Sampling Distribution of p Assumptions 1 Simple random sample from the population Sampling Distribution of p Assumptions 1 Simple random sample from the population 2 A large enough sample so that the central limit theorem applies The sample is large if both the expected number of successes np and the expected number of failures n 1 p are greater than or equal to 10 Sampling Distribution of p Assumptions 1 Simple random sample from the population 2 A large enough sample so that the central limit theorem applies The sample is large if both np and n 1 p are greater than or equal to 10 Then the sample proportion p is distributed approximately normal with mean m p p and standard deviation sp p 1 p n p N p p 1 p n Z Score Transformation Z p p has a standard normal Z distribution p 1 p n Confidence Interval Potential expression for a 100 C confidence interval for p p z p 1 p n Confidence Interval Potential expression for a 100 C confidence interval for p p Z p 1 p n However p is unknown Confidence Interval Potential expression for a 100 C confidence interval for p p Z p 1 p n However p is unknown We substitute the sample proportion p for p A similar substitution is made with the assumptions and hence the assumptions and confidence interval formula become Confidence Interval Assumptions 1 Simple random sample from the population 2 Large sample quantified as both the number of successes in the sample np and the number of failures in the sample n 1 p are both greater than or equal to 10 Then a 100 C confidence interval for p is p z p 1 p n Confidence Interval Assumptions 1 Simple random sample from the population 2 Large sample quantified as both the number of successes in the sample np and the number of failures in the sample n 1 p are both greater than or equal to 10 Then a 100 C confidence interval for p is p z p 1 p n The z values are found in the table on page 340 TI 83 84 Calculator See pages 251 252 or 216 217 for instructions on using a calculator to determine confidence intervals for proportions 1 Hit STAT 2 Scroll over to TESTS 3 Choose A 1 PropZInt 4 X is the number of successes n is the sample size 5 Enter required components and hit Calculate Example 89 71 Population of interest Parameter of interest Example 89 71 Population of interest all current college presidents Parameter of interest p proportion of all current college presidents who favor the plan Example 89 71 Simple random sample of n 100 college presidents p number of successes n Example 89 71 Simple random sample of n 100 college presidents p number of successes 6 06 n 100 Example 89 71 Assumptions 1 Do we have a simple random sample Example 89 71 Assumptions 1 Do we have a simple random sample YES Example 89 71 Assumptions 1 Do we have a simple random sample YES 2 Do we have a large enough sample for the CLT to apply Example 89 71 Assumptions 1 Do we have a simple random sample YES 2 Do we have a large enough sample for the CLT to apply NO the number of successes in the sample is only 6 which is less than 10 Hence the assumptions are not satisfied and the confidence interval should not be calculated Example 90 72 Population of interest Parameter of interest Example 90 72 Population of interest all current athletic directors Parameter of interest p proportion of all athletic directors who favor the plan Example 90 72 Simple random sample of n 100 athletic directors p number of successes Top Hat n Example 90 72 Simple random sample of n 100 athletic directors p number of successes 25 25 n 100 Example 90 72 Simple random sample of n 100 athletic directors p number of successes 25 25 n 100 np 100 25 25 and n 1 p 100 1 25 75 Assumptions 1 Simple random sample 2 Large enough sample for the CLT to apply

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