VCU STAT 210  Lecture29 (26 pages)
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Lecture29
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 Stat 210  Basic Practice of Statistics
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STAT 210 Lecture 29 Inferences on Population Proportions November 3 2017 Test 5 Monday November 6 Covers chapter on proportions and concepts of 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 1 through IX 18 Recommended problem IX 15 Hummingbird Pages 210 through 215 Relevant problems VIII 1 through VIII 18 Recommended problem VIII 15 Additional Reading and Examples Sailboat Read pages 248 through 251 Hummingbird Read pages 206 through 209 Statistical Inference Statistical inference involves using statistics computed from a sample data to make statements about some parameter of the population This chapter we have made inferences about the population proportion p Top Hat 2 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 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 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 Assumptions Simple random sample 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 Confidence Interval 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 Properties 1 Increase the degree of confidence then the margin of error and hence the width of the interval increase Decrease the degree of confidence then the margin of error and hence the width of the interval decrease 2 Increase the sample size then the margin of error and hence the width of the interval decrease Decrease the sample size then the margin of error and hence the width of the interval increase Significance Test for p We hypothesize that the population proportion p equals some specified value p0 and we want to use the data in a sample to test whether this null hypothesis is appropriate or whether we should reject the null hypothesis in favor of some alternative hypothesis Null hypothesis H0 p p 0 Ha p p 0 Alternative hypothesis Ha p p 0 Ha p p 0 Significance Test for p Assumptions Simple random sample Both the hypothesized number of successes np0 and the hypothesized number of failures n 1 p0 are both greater than or equal to 10 Z p p0 p0 1 p0 n Top Hat Card Exercise 1 Half the cards in the deck are red and half the cards in the deck are black so proportion of cards that are red is 50 Card Exercise 1 Half the cards in the deck are red and half the cards in the deck are black so proportion of cards that are red is 50 2 H0 p 50 versus HA p 50 Card Exercise 1 Half the cards in the deck are red and half the cards in the deck are black so proportion of cards that are red is 50 2 H0 p 50 versus HA p 50 3 np 25 50 12 5 red cards and likewise n 1 p 25 1 50 12 5 black cards Card Exercise 4 Since we have a simple random sample and both np and n 1 p are greater than 10 then the assumptions are satisfied mp p 50 sp p 1 p n 50 1 50 10 25 Shape is approximately normal Card Exercise Number of red cards chosen Number of black cards chosen p 25 Card Exercise 6 Does this data provide some evidence that the alternative hypothesis HA p 50 is true Top Hat Card Exercise 7 1 H0 p 50 versus HA p 50 test at a 10 2 Assumptions satisfied question 4 p Z 50 50 1 50 10 25 3 p value 2P Z 2P Z 2 1 P Z 2 1 2 Card Exercise 4 a 10 Since p value 10 we 5 There is sufficient insufficient evidence that the proportion of cards in the deck that are red is different from 50 Card Exercise We concluded that there is not sufficient evidence that the proportion of cards in the deck that are red is different from 50 Is this the correct conclusion Top Hat 2 Card Exercise Number of red cards chosen Number of black cards chosen p 25 If appropriate use these results to calculate and interpret a 90 confidence interval for the proportion of red cards in this deck Card Exercise Confidence Interval n 25 p 25 We have a simple random sample and np and n 1 p are greater than 10 For a 90 confidence interval z 1 645 1 645 1 25 We have 90 confidence that the proportion of all cards in the deck that are red is between and
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