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Chapter 7

Chapter 7. Estimation Procedures. Chapter Outline. Introduction Bias and Efficiency Estimation Procedures: Introduction Interval Estimation Procedures for Sample Means (Large Samples) Interval Estimation Procedures for Sample Proportions (Large Samples). Chapter Outline.

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Chapter 7

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  1. Chapter 7 Estimation Procedures

  2. Chapter Outline • Introduction • Bias and Efficiency • Estimation Procedures: Introduction • Interval Estimation Procedures for Sample Means (Large Samples) • Interval Estimation Procedures for Sample Proportions (Large Samples)

  3. Chapter Outline • A Summary of the Computation of Confidence Intervals • Controlling the Width of Interval Estimates • Interpreting Statistics: Predicting the Election of the President and Judging His Performance

  4. In This Presentation • The logic of estimation • How to construct and interpret interval estimates for: • Sample means • Sample Proportions

  5. Basic Logic • In estimation procedures, statistics calculated from random samples are used to estimate the value of population parameters. • Example: • If we know 42% of a random sample drawn from a city are Republicans, we can estimate the percentage of all city residents who are Republicans.

  6. Basic Logic • Information from samples is used to estimate information about the population. • Statistics are used to estimate parameters. POPULATION SAMPLE PARAMETER STATISTIC

  7. Sampling Distribution is the link between sample and population. The value of the parameters are unknown but characteristics of the S.D. are defined by theorems. Basic Logic POPULATION SAMPLING DISTRIBUTION SAMPLE

  8. Two Estimation Procedures • A point estimate is a sample statistic used to estimate a population value. • A newspaper story reports that 74% of a sample of randomly selected Americans support capital punishment. • Confidence intervals consist of a range of values. • ”between 71% and 77% of Americans approve of capital punishment.”

  9. Constructing Confidence Intervals For Means • Set the alpha (probability that the interval will be wrong). • Setting alpha equal to 0.05, a 95% confidence level, means the researcher is willing to be wrong 5% of the time. • Find the Z score associated with alpha. • If alpha is equal to 0.05, we would place half (0.025) of this probability in the lower tail and half in the upper tail of the distribution. • Substitute values into equation 7.3.

  10. Confidence Intervals For Means: Problem 7.5c • For a random sample of 178 households, average TV viewing was 6 hours/day with s = 3. Alpha = .05. • c.i. = 6.0 ±1.96(3/√177) • c.i. = 6.0 ±1.96(3/13.30) • c.i. = 6.0 ±1.96(.23) • c.i. = 6.0 ± .44

  11. Confidence Intervals For Means • We can estimate that households in this community average 6.0±.44 hours of TV watching each day. • Another way to state the interval: • 5.56≤μ≤6.44 • We estimate that the population mean is greater than or equal to 5.56 and less than or equal to 6.44. • This interval has a .05 chance of being wrong.

  12. Confidence Intervals For Means • Even if the statistic is as much as ±1.96 standard deviations from the mean of the sampling distribution the confidence interval will still include the value of μ. • Only rarely (5 times out of 100) will the interval not include μ.

  13. Constructing Confidence Intervals For Proportions • Procedures: • Set alpha. • Find the associated Z score. • Substitute the sample information into Formula 7.3.

  14. Confidence Intervals For Proportions • If 42% of a random sample of 764 from a Midwestern city are Republicans, what % of the entire city are Republicans? • Don’t forget to change the % to a proportion. • c.i. = .42 ±1.96 (√.25/764) • c.i. = .42 ±1.96 (√.00033) • c.i. = .42 ±1.96 (.018) • c.i. = .42 ±.04

  15. Confidence Intervals For Proportions • Changing back to %s, we can estimate that 42% ± 4% of city residents are Republicans. • Another way to state the interval: • 38%≤Pu≤ 46% • We estimate the population value is greater than or equal to 38% and less than or equal to 46%. • This interval has a .05 chance of being wrong.

  16. SUMMARY • In this situation, identify the following: • Population • Sample • Statistic • Parameter

  17. SUMMARY • Population = All residents of the city. • Sample = the 764 people selected for the sample and interviewed. • Statistic = Ps = .42 (or 42%) • Parameter = unknown. The % of all residents of the city who are Republican.

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