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Confidence Interval Estimation for a Population Proportion

Learn about point estimates and the benefits of interval estimates in estimating population proportions. Discover how to calculate a 95% confidence interval and interpret the results. Explore different confidence levels and the probability of error associated with each level.

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Confidence Interval Estimation for a Population Proportion

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  1. Confidence Interval Estimation for a Population Proportion Lecture 33 Section 9.4 Tue, Mar 22, 2005

  2. Point Estimates • Point estimate – A single value of the statistic used to estimate the parameter. • The problem with point estimates is that we have no idea how close we can expect them to be to the parameter. • That is, we have no idea of how large the error may be.

  3. Interval Estimates • Interval estimate – an interval of numbers that has a stated probability (often 95%) of containing the parameter. • An interval estimate is more informative than a point estimate.

  4. Interval Estimates • Confidence level – The probability that is associated with the interval. • If the confidence level is 95%, then the interval is called a 95% confidence interval.

  5. Approximate 95% Confidence Intervals • How do we find a 95% confidence interval for p? • Begin with the sample size n and the sampling distribution of p^. • We know that the sampling distribution is normal with mean p and standard deviation

  6. Approximate 95% Confidence Intervals • Therefore… • Approximately 95% of all values of p^ are within 2 standard deviations of p. • Therefore… • For a single random p^, there is a 95% chance that it is within 2 standard deviations of p. • Therefore… • There is a 95% chance that p is within 2 standard deviations of a single random p^.

  7. Approximate 95% Confidence Intervals • Thus, the confidence interval is • The trouble is, to know p^, we must know p. (See the formula for p^.) • The best we can do is to use p^ in place of p to estimate p^.

  8. Approximate 95% Confidence Intervals • That is, • This is called the standard error of p^ and is denoted SE(p^). • Now the 95% confidence interval is

  9. Example • Example 9.6, p. 539. • The answer is (0.178, 0.206). • That means that we are 95% confident, or sure, that p is somewhere between 0.178 and 0.206.

  10. Let’s Do It! • Let’s do it! 9.5, p. 540 – When Do You Turn Off Your Cell Phone?

  11. Confidence Intervals • We are using the number 2 as a rough approximation for a 95% confidence interval. • We can get a more precise answer if we use the normal tables. • A 95% confidence interval cuts off the upper 2.5% and the lower 2.5%. • What values of z do that?

  12. Standard Confidence Levels • The standard confidence levels are 90%, 95%, 99%, and 99.9%. (See p. 542.)

  13. The Confidence Interval • The confidence interval is given by the formula where z is given by the previous chart or is found in the normal table.

  14. Confidence Level • Rework Let’s Do It! 9.5, p. 540, by computing a • 95% confidence interval. • 90% confidence interval. • 99% confidence interval. • Which one is widest? • Which one is best?

  15. Probability of Error • We use the symbol  to represent the probability that the confidence interval is in error. • That is,  is the probability that p is not in the confidence interval. • In a 95% confidence interval,  = 0.05.

  16. Probability of Error • Thus, the area in each tail is /2. • The value of z can be found by using the invNorm function on the TI-83. • For example, • 90% CI:  =0.10; invNorm(0.05) = –1.645. • 95% CI:  =0.05; invNorm(0.025) = –1.960. • 99% CI:  =0.01; invNorm(0.005) = –2.576. • 99.9% CI:  =0.001; invNorm(0.0005) = –3.291.

  17. Values of z

  18. Think About It • Think about it, p. 542. • Which is better? • A wider confidence interval, or • A narrower confidence interval. • Which is better? • A low level of confidence, or • A high level of confidence.

  19. Think About It • Which is better? • A smaller sample, or • A larger sample. • What do we mean by “better”? • Is it possible to increase the level of confidence and make the confidence narrower at the same time?

  20. TI-83 – Confidence Intervals • The TI-83 will compute a confidence interval for a population proportion. • Press STAT. • Select TESTS. • Select 1-PropZInt.

  21. TI-83 – Confidence Intervals • A display appears requesting information. • Enter x, the numerator of the sample proportion. • Enter n, the sample size. • Enter the confidence level, as a decimal. • Select Calculate and press ENTER.

  22. TI-83 – Confidence Intervals • A display appears with several items. • The title “1-PropZInt.” • The confidence interval, in interval notation. • The sample proportion p^. • The sample size. • How would you find the margin of error?

  23. TI-83 – Confidence Intervals • Rework Let’s Do It! 9.5, p. 540, using the TI-83.

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