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Section 8.2 Estimating a Population Proportion

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Section 8.2 Estimating a Population Proportion After this section, you should be able to… CONSTRUCT and INTERPRET a confidence interval for a population proportion DETERMINE the sample size required to obtain a level C confidence interval for a population proportion with a specified margin of error DESCRIBE how the margin of error of a confidence interval changes with the sample size and the level of confidence C

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Estimating p & Constructing a CI: Conditions 1) Random: The data should come from a well-designed random sample or randomized experiment. 3) Independent: Individual observations are independent. When sampling without replacement, the sample size n should be no more than 10% of the population size N (the 10% condition).

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Constructing the Formula for CI for p

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The Process: Confidence Intervals Check Conditions: Random: SRS Normal: be sure to show both np ≥ 10 and n (1 – p ) ≥ 10 and the formula plugged in as part of checking this condition.Independent: 10% rule Perform Calculations: You can use your calculator. “Z Interval” Conclude in Context: Be sure to answer the question being asked.

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Example: Red Beads Calculate and interpret a 90% confidence interval for theproportion of red beads in the container of 3000 beads. Yourteacher claims 50% of the beads are red. Use your interval tocomment on this claim. The sample proportion you foundwas 0.426 with a sample of 251 beads. z.03.04.05 – 1.7.0418.0409.0401 – 1.6.0516.0505.0495 – 1.5.0630.0618.0606 For a 90% confidence level, z* = 1.645

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Example: Red Beads Calculate and interpret a 90% confidence interval for the proportion of red beads inthe container of 3000 beads. Your teacher claims 50% of the beads are red. Use yourinterval to comment on this claim. The sample proportion you found was 0.426 with asample of 251 beads. For a 90% confidence level, z* = 1.645 statistic ± (critical value) (standard deviation of the statistic) 0.426 +/- 1.645*

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Example: Red Beads Calculate and interpret a 90% confidence interval for the proportion of red beads inthe container of 3000 beads. Your teacher claims 50% of the beads are red. Use yourinterval to comment on this claim. The sample proportion you found was 0.426 witha sample of 251 beads. For a 90% confidence level, z* = 1.645 statistic ± (critical value) (standard deviation of the statistic) 0.426 +/- 1.645*

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Example: Red Beads Calculate and interpret a 90% confidence interval for the proportion of red beads inthe container of 3000 beads. Your teacher claims 50% of the beads are red. Use yourinterval to comment on this claim. The sample proportion you found was 0.426 witha sample of 251 beads. We are 90% confident that the true population proportion of red beads is between 0.375 and 0.477. Therefore the teacher’s claim that 50% of the beads are red is doubtful because 0.50 is not included within the interval.

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Example: Customer Satisfaction MacDonald's wants to determine customer satisfaction with itsnew BBQ Chicken Burger. The VP of New Products has hiredyou to conduct a survey. At a minimum how many people doyou need to survey, if the company is requiring a margin oferror of 0.03 at 95% confidence?

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Example: Customer Satisfaction MacDonald's wants to determine customer satisfaction with itsnew BBQ Chicken Burger. The VP of New Products has hired youto conduct a survey. At a minimum how many people do you needto survey, if the company is requiring a margin of error of 0.03 at95% confidence? Multiply both sides by square root n and divide both sides by 0.03. Square both sides. Substitute 0.5 for the sample proportion to find the largest ME possible. We round up to 1068 respondents to ensure the margin of error is no more than 0.03 at 95% confidence.

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