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Introduction to Confidence Intervals using Population Parameters

Introduction to Confidence Intervals using Population Parameters. Chapter 10.1 & 10.3. Rate your confidence 0 (no confidence) – 100 (very confident). Name my age within 10 years? within 5 years? within 1 year ? What happens to your confidence as the interval (age range) gets smaller?.

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Introduction to Confidence Intervals using Population Parameters

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  1. Introduction to Confidence Intervals using Population Parameters Chapter 10.1 & 10.3

  2. Rate your confidence0 (no confidence) – 100 (very confident) • Name my age within 10 years? • within 5 years? • within 1 year? • What happens to your confidence as the interval (age range) gets smaller?

  3. Would you agree? As my age interval decreases your confidence decreases. On the other hand, your confidence increases as the interval widens, because you are given a greater margin of error.

  4. Point Estimate • When we use a single statistic based on sample data to estimate a population parameter • Simplest approach • But not always very precise due to variation in the sampling distribution

  5. Confidence intervals • Are used to estimate the unknown population parameter • Formula: estimate + margin of error

  6. Margin of error • Shows how accurate we believe our estimate is • The smaller the margin of error, m, the more precise our estimate of the true parameter. • Formula:

  7. Confidence level • Is the success rate of the methodused to construct the interval. • Using this method, ____% of the intervals constructed will contain the true population parameter.

  8. What does it mean to be 95% confident? • 95% chance that the true p is contained in the confidence interval • The probability that the interval contains the true p is 95% • The method used to construct the interval will produce intervals that contain the true p 95% of the time.

  9. Critical value (z*) • Found from the confidence level • The upper z-score with probability p lying to its right under the standard normal curve Confidence level(C) each tail area z* 90% .10/2 =.05 1.645 95% .05/2 =.025 1.96 99% .01/2 =.005 2.576 • z* can be looked up in table or, by using • 2nd VARS #3 invNorm(1.C/2 = • Example: 2nd VARS #3 invNorm(1.95/2 = 2.575829

  10. Confidence interval for a population proportion:

  11. Steps for doing a confidence interval: • State the parameter of interest. • Name inference procedure & state assumptions. • See assumptions for CI for population parameter on next slide. • Calculate the confidence interval using formula. • Write a statement about the interval in the context of the problem.

  12. CI assumptions for a pop. parameter Step 2: Name inference procedure and state assumptions: • SRS from population 2) Normality: The number of success and failures are both at least 10. • Note: On AP Test you must show the calculation below, simply stating the number of successes and failures are both at least 10 isn’t enough. • np> 10 & n(1-p) > 10. 3) Independence: Population size > 10n

  13. Statement: (memorize!!) We are ________% confident that the true proportion context lies within the interval ______ and ______.

  14. Your local newspaper polls a random sample of 330voters, finding 144who say they will vote “yes” on the upcoming school budget. Create a 95 % confidence interval for actual sentiment of all voters. 1st Calculatep-hat = 144/330 = .436 • Assumptions: • The voters were sampled randomly. • 330(.436)=144 & 330(.564)=186, both ≥ 10 • Population of eligible voters must be at least 3300 = 10(330). • We are 95% confident that the true proportion of voters that will vote “yes” is between .382 and .490.

  15. An experiment finds that 27% of 53 randomly sampled subjects report improvement after using a new medicine. Create a 95% confidence interval for the actual cure rate. • Assumptions: • The subjects were sampled randomly • 53 (.27)=14 and 53(.73)=39, both ≥10 • The population of subjects using this new medicine must be at least 530 = 10(53) • We are 95% confident that the true proportion of people that will improve after using the new medication is between .15 and .39.

  16. 90% confidence interval? • We are 90% confident that the true proportion of people that will improve after using the new medication is between .17 and .37.

  17. How can you make the margin of error smaller? • z* smaller (lower confidence level) • s smaller (less variation in the population) • n larger (to cut the margin of error in half, n, the sample size must be 4 times as big) In real life, you can’t adjust

  18. If a certain margin of error is wanted, then to find the sample size necessary for that margin of error use: Find a sample size: Always round up to the nearest person/object!

  19. Find the sample size required for ±5%, with 98% confidence. Consider the formula for margin of error. We believe the improvement rate to be .27 from our preliminary study. We need to run an experiment with at least 427 people receiving the new medication in order to have a margin of error of ±5%, with 98% confidence.

  20. Use 95% confidence and .5 for p-hat When they don’t give you a % of confidence or p-hat:

  21. What sample size does it take to estimate the outcome for an election with a margin of error of 3%? We need to have a sample size of at least 1068 people to estimate the outcome for an election in order to have a margin of error of ±3%, with 95% confidence.

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