1. Section 7.3 Confidence intervals for a population proportion
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Section 7.3 Confidence intervals for a population proportion
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2. Objectives Construct a confidence interval for a population proportion
Find the sample size necessary to obtain a confidence interval of a given width
Describe a method for constructing confidence intervals with small samples
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3. Objective 1
Construct a confidence interval for a population proportion
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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.
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5. The point estimate for the population proportion is 𝑝 = 𝑥 𝑛 .
Notation
We use the following notation:
p is the population proportion of individuals who are in a specified category.
x is the number of individuals in the sample who are in the specified category.
n is the sample size.
𝑝 is the sample proportion of individuals who are in the specified category.
The point estimate for the population proportion is 𝑝 = 𝑥 𝑛 .
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6. Standard Error and Margin of Error
The standard error is determined by the sampling distribution of 𝑝 and is given by: Standard Error = 𝒑 (𝟏− 𝒑 ) 𝒏 The margin of error is computed as the critical value 𝑧 𝛼/2 times the standard error: Margin of Error = 𝒛 𝜶 𝟐 ∙ 𝒑 (𝟏− 𝒑 ) 𝒏
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7. Confidence Interval for Population Proportion p
The confidence interval for the population proportion p is
Point estimate ± Margin of error
𝒑 ± 𝒛 𝜶 𝟐 ∙ 𝒑 (𝟏− 𝒑 ) 𝒏
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8. Assumptions for Constructing Confidence Interval for p
Assumptions for Constructing a Confidence Interval for p
We have a simple random sample.
The population is at least 20 times as large as the sample.
The items in the population are divided into two categories.
The sample must contain at least 10 individuals in each category.
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9. Procedure for Constructing a Confidence Interval for p
Check to be sure the assumptions are satisfied.
Step 1: Compute the value of the point estimate 𝑝 .
Step 2: Find the critical value 𝑧 𝛼 2 for the desired confidence level.
Step 3: Compute the standard error 𝑝 (1− 𝑝 ) 𝑛 and multiply it by the
critical value to obtain the margin of error 𝑧 𝛼 𝑝 (1− 𝑝 ) 𝑛 .
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10. Procedure for Constructing a Confidence Interval for p
Step 4: Use the point estimate and the margin of error to construct the
confidence interval:
Point estimate ± Margin of error
𝑝 ± 𝑧 𝛼 2 ∙ 𝑝 (1− 𝑝 ) 𝑛
𝑝 − 𝑧 𝛼 2 ∙ 𝑝 (1− 𝑝 ) 𝑛 < p < 𝑝 + 𝑧 𝛼 2 ∙ 𝑝 (1− 𝑝 ) 𝑛
Step 5: Interpret the result.
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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.
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12. Example
Solution (continued): Step 1: Compute the point estimate 𝒑 . Since the sample size is n = 517 and the number who believe that video games have a positive effect is x = 403, the point estimate is 𝑝 = 𝑥 𝑛 = = Step 2: Find the critical value. For a 95% confidence interval, the critical value is 𝑧 𝛼 2 =1.96, found at the bottom of Table A.3 or by technology.
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13. Example
Solution (continued): Step 3: Compute the margin of error. The margin of error is 𝑧 𝛼 2 ∙ 𝑝 (1− 𝑝 ) 𝑛 = 1.96 ∙ (1 − ) 517 = Step 4: Construct the confidence interval: The confidence interval is 𝑝 − 𝑧 𝛼 2 ∙ 𝑝 (1− 𝑝 ) 𝑛 < p < 𝑝 + 𝑧 𝛼 2 ∙ 𝑝 (1− 𝑝 ) 𝑛 – < p < < p < 0.815
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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.
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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.
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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).
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17. Objective 2
Find the sample size necessary to obtain a confidence interval of a given width
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18. Reducing the Margin of Error
If we wish to make the margin of error of a confidence interval smaller while keeping the confidence level the same, we can do this by making the sample size larger. Sometimes we have a specific value 𝑚 that we would like the margin of error to attain, and we wish to compute a sample size 𝑛 that is likely to give us a margin of error of that size. Let m represent the margin of error: 𝑚 = 𝑧 𝛼 2 ∙ 𝑝 (1− 𝑝 ) 𝑛 This formula may be rewritten as 𝑛= 𝑝 (1− 𝑝 ) 𝑧 𝛼 2 𝑚 2 .
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19. Necessary Sample Size Formula
The formula for the required sample size is
In order to use this formula, we need a value for 𝑚 and 𝑝 . We can set the value of 𝑚, but we don’t know ahead of time what 𝑝 is going to be.
There are two ways to determine a value for 𝑝 .
Use a value that is available from a previously drawn sample.
To assume that 𝑝 = 0.5, which makes the margin of error as large as possible for any sample size. In this case, the formula simplifies to 𝑛= 𝑧 𝛼 2 𝑚 2 .
If the value of n given by the formula is not a whole number, round up to the nearest whole number.
𝑛= 𝑝 (1− 𝑝 ) 𝑧 𝛼 2 𝑚 2
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20. 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. Estimate the sample size needed so that a 95% confidence interval will have a margin of error of Solution: Since the desired level is 95%, the critical value is 𝑧 𝛼 2 = We compute 𝑝 = 𝑥 𝑛 = = The desired margin of error is 𝑚 = The necessary sample size is 𝑛= 𝑝 (1− 𝑝 ) 𝑧 𝛼 2 𝑚 2 = ( )(1 – ) =
We round up to 734.
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21. Example
We plan to sample music teachers in order to construct a 95% confidence interval for the proportion who believe that listening to hip-hop music has a positive effect on music education. We have no value of 𝑝 available. Estimate the sample size needed so that a 95% confidence interval will have a margin of error of Solution: The critical value is therefore 𝑧 𝛼 2 = 1.96 and the desired margin of error is m = Since we have no value of 𝑝 , substitute the values 𝑧 𝛼 2 = 1.96 and m = 0.03 into the formula 𝑛=0.25 𝑧 𝛼 2 𝑚 2 . We have 𝑛= = We round up to 1068.
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22. Objective 3
Describe a method for constructing confidence intervals with small samples
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23. Adjusted Sample Proportion 𝑝
The method presented for constructing a confidence interval for a proportion requires that we have at least 10 individuals in each category. When this condition is not met, we can still construct a confidence interval by adjusting the sample proportion a bit. We increase the number of individuals in each category by 2, so that the sample size increases by 4. Thus, instead of using the sample proportion 𝑝 = 𝑥 𝑛 , we use the adjusted sample proportion, 𝑝 .
Adjusted sample proportion
𝑝 = 𝑥+2 𝑛+4
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24. Standard error and Critical values for a Proportion with Small Samples
The standard error and critical value are calculated in the same way as in the traditional method, except that we use the adjusted sample proportion 𝑝 in place of 𝑝 , and n + 4 in place of n. The standard error becomes 𝑝 (1− 𝑝 ) 𝑛+4 .
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25. Example
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: The adjusted sample proportion is 𝑝 = 𝑥+2 𝑛+4 = = The critical value is 𝑧 𝛼 2 = The confidence interval is 𝑝 – 𝑧 𝛼 2 𝑝 (1− 𝑝 ) 𝑛+4 < p < 𝑝 + 𝑧 𝛼 2 𝑝 (1− 𝑝 ) 𝑛 – (1−0.5714) < p < (1−0.5714) < p < 0.831
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26. Small-Sample Method: TI-84 PLUS
Because the only difference between the small-sample method and the traditional method is the use of 𝑝 rather than 𝑝 , a software package or calculator such as TI-84 Plus can be made to produce a confidence interval using the small-sample method. Simply input x + 2 for the number of individuals in the category of interest, and n + 4 for the sample size.
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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).
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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;
The margin of error is smaller, because we divide by n + 4 rather than n.
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.
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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?
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