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Estimating with Confidence

Learn how to interpret confidence levels and intervals in context and understand the importance of random, normal, and independent conditions in estimating a parameter.

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Estimating with Confidence

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  1. Ch 8 • Estimating with Confidence

  2. Today’s Objectives • I can interpret a confidence level. • I can interpret a confidence interval in context. • I can understand that a confidence interval gives a range of plausible values for the parameter.

  3. The Basic Idea • We will use the formulas and ideas from the previous chapter. • This time we will not know the true population parameters. -- This is what really happens. • Inference is what Statistics is all about!

  4. Vocabulary • Point Estimator - what we’re using • Point Estimate - the value of what we’re using • The best estimator has no bias and low variability!

  5. Determine the point estimator you would use and calculate the value of the point estimate: • a) The makers of a new golf ball want to estimate the median distance the new balls will travel when hit by a mechanical driver. They select a random sample of 10 balls and measure the distance each ball travels. Here are the distances (in yards): • 285 266 284 285 282 284 287 290 288 285 • b) The math department wants to know what proportion of its students who own a graphing calculator, so they take a random sample of 100 students and find that 28 own a graphing calculator.

  6. How can we find the confidence interval of our mystery mean? • Remember, if the population is Normal then so is the sampling distribution. • and 10% condition is met because we are sampling from an infinite population in this case.

  7. How can we find the confidence interval of our mystery mean? • About how many standard deviation from the mean represents a 95% interval? Therefore, we can estimate that the real mean lies somewhere in this interval: This happens in about 95% of all possible samples.

  8. Confidence Interval for a parameter:estimate + margin of error • Margin of Error: how close the estimate tends to be to the unknown parameter in repeated samples • accounts for the variability due to random selection. It does NOT compensate for bias in data collection! • Three ways to write it

  9. Confidence Level, C • * gives the overall success rate of the method for calculating the confidence interval. • * That is, in C% of all possible samples, the method would yield an interval that captures the true parameter value. • * let’s look at figure 8.5 on page 474

  10. Confidence Interval Visual

  11. Interpreting Confidence Level and Confidence interval • Confidence Level: C% of all possible samples of a given size from this population will result in an interval that captures the unknown parameter. • Confidence Interval: I am C% confident that the interval from __ to __ captures the actual value of the [population parameter in context]

  12. Don’t confuse confidence levels with confidence intervals! • Always interpret the interval, only interpret the level when asked • Confidence level does NOT tell us the chance that an interval captures the population parameter! • It gives us a set of plausible values

  13. What are the confidence level and confidence interval of our mystery mean this time? • Also, interpret them. **What is the probability that our mystery mean is in our confidence interval?

  14. According to www.gallup.com on August 13, 2010, the 95% confidence interval for the true proportion of Americans who approved of the job President Obama was doing was 0.44 + 0.03 • Interpret the confidence level and confidence interval.

  15. 8.1 Day 2 • Objectives: • I can understand why each of the 3 conditions are important. • I can explain how practical issues affect the interpretation of a confidence interval.

  16. Confidence Interval: • statistic + (critical value) (standard deviation of statistic) (this is on your formula sheet) The critical value depends on both, the confidence level, C, and the sampling distribution of the statistic.

  17. How to get a small margin of error? • Lower your confidence level • Increase your sample size WHY? Can you think of pros and cons of each?

  18. Conditions to check before calculating a confidence interval: • Random • Normal • Independent Read page 479 about these in detail!

  19. Random • Randomization must be used correctly! • If not, our estimates might be biased and we shouldn’t have any confidence that the intervals we calculate will actually contain the value of the parameter we’re trying to estimate.

  20. Normal • all calculations depend on the fact that the sampling distribution is approximately Normal! • We will be over confident is we assume Normality when the population is in reality skewed. • Remember there are two different tests for Normality and it depends on if we are using sample means or sample proportions!

  21. Independent • our standard deviation formulas depend on replacing the sample; rarely is that the case. • This makes checking the 10% condition more critical!

  22. Why these 3 conditions? • They each are related to the 3 parts of the formula: • Random ensures the statistic is unbiased. • Normal ensures that we are using the correct critical value • Independence ensures we are using the correct formula for the standard deviation of the statistic

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