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SECTION 7.3 CONFIDENCE INTERVALS FOR A POPULATION PROPORTION Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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Presentation on theme: "SECTION 7.3 CONFIDENCE INTERVALS FOR A POPULATION PROPORTION Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display."— Presentation transcript:

1 SECTION 7.3 CONFIDENCE INTERVALS FOR A POPULATION PROPORTION Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

2 Objectives 1. Construct a confidence interval for a population proportion 2. Find the sample size necessary to obtain a confidence interval of a given width 3. Describe a method for constructing confidence intervals with small samples Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

3 Construct a confidence interval for a population proportion Objective 1 Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4 Guitar Hero The music organization Little Kids Rock surveyed 517 music teachers, and 403 of them said that video games like Guitar Hero and Rock Band, in which 442 players try to play music in time with a video image, have a positive effect on music education. Assuming these teachers to be a random sample of U.S. music teachers, we would like to construct a confidence interval for the proportion of music teachers who believe that music video games have a positive effect on music classrooms. Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

5 Notation Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

6 Standard Error and Margin of Error Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

7 Confidence Interval for Population Proportion p Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

8 Assumptions for Constructing Confidence Interval for p Assumptions for Constructing a Confidence Interval for p 1.We have a simple random sample. 2.The population is at least 20 times as large as the sample. 3.The items in the population are divided into two categories. 4.The sample must contain at least 10 individuals in each category. Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

9 Procedure for Constructing a Confidence Interval for p Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

10 Procedure for Constructing a Confidence Interval for p Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

11 Example In a survey of 517 music teachers, 403 said that the video games Guitar Hero and Rock Band have a positive effect on music education. Construct a 95% confidence interval for the proportion of music teachers who believe that these video games have a positive effect. Solution: We first check the assumptions: We have a simple random sample. It is reasonable to believe that the population of music teachers in the U.S. is at least 20 times as large as the sample. The items in the population can be divided into two categories: those who believe that the games have a positive effect, and those who do not. There are 403 teachers who believe that the games have a positive effect, and 517 − 403 = 114 who do not, so there are 10 or more items in each category. The assumptions are met. Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

12 Example Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

13 Example Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

14 Solution Solution (continued): Step 5: Interpret the result. We are 95% confident that the proportion of music teachers who believe that video games have a positive effect is between and Suppose that a video game manufacturer claims that 80% of music teachers believe that Guitar Hero and Rock Band have a positive effect. Does the confidence interval contradict this claim? Because the value 0.80 is within the confidence interval, the confidence interval does not contradict the claim. Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

15 Confidence Intervals on the TI-84 PLUS The 1-PropZInt command constructs confidence intervals for the population proportion. This command is accessed by pressing STAT and highlighting the TESTS menu. Enter the values of x, n, and the confidence level. Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

16 Example (TI-84 PLUS) In a survey of 517 music teachers, 403 said that the video games Guitar Hero and Rock Band have a positive effect on music education. Construct a 95% confidence interval for the proportion of music teachers who believe that these video games have a positive effect. Solution: We press STAT and highlight the TESTS menu and select 1-PropZInt. Enter 403 in the x field, 517 in the n field, and 0.95 in the C-level field. Select Calculate. The confidence interval is (0.744, 0.815). Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

17 Find the sample size necessary to obtain a confidence interval of a given width Objective 2 Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

18 Reducing the Margin of Error Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

19 Necessary Sample Size Formula Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

20 Example We round up to 734. Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

21 Example Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

22 Describe a method for constructing confidence intervals with small samples Objective 3 Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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24 Standard error and Critical values for a Proportion with Small Samples Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

25 Example Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

26 Small-Sample Method: TI-84 PLUS Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

27 Example (TI-84 PLUS) In a random sample of 10 businesses in a certain city, 6 of them had more than 15 employees. Use the small-sample method to construct a 95% confidence interval for the proportion of businesses in this city that have more than 15 employees. Solution: We press STAT and highlight the TESTS menu and select 1-PropZInt. Enter 8 (which is 6+2) in the x field, 14 (which is 10+4) in the n field, and 0.95 in the C-level field. Select Calculate. The confidence interval is (0.312, 0.831). Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

28 Advantages of the Small-Sample Method The small-sample method can be used for any sample size, and recent research has shown that it has two advantages over the traditional method; 1. The margin of error is smaller, because we divide by n + 4 rather than n. 2. The actual probability that the small-sample confidence interval covers the true proportion is almost always at least as great as, or greater than, that of the traditional method. Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

29 Do You Know… How to construct a confidence interval for a population proportion? How to find the sample size necessary to obtain a confidence interval for a population proportion of a given width? How to construct confidence intervals with small samples? Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.


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