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Understanding Point Estimates and Confidence Intervals

Learn about point estimates, errors of estimation, confidence levels, critical values, and creating confidence intervals for population parameters based on sample data.

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Understanding Point Estimates and Confidence Intervals

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  1. Point Estimate an estimate of a population parameter given by a single number

  2. Examples of Point Estimates

  3. Examples of Point Estimates • is used as a point estimate for  .

  4. Examples of Point Estimates • is used as a point estimate for  . • s is used as a point estimate for .

  5. Error of Estimate the magnitude of the difference between the point estimate and the true parameter value

  6. The error of estimate using as a point estimate for  is

  7. Confidence Level • A confidence level, c, is a measure of the degree of assurance we have in our results. • The value of c may be any number between zero and one. • Typical values for c include 0.90, 0.95, and 0.99.

  8. Critical Value for a Confidence Level, c the value zc such that the area under the standard normal curve falling between – zc and zc is equal to c.

  9. – zc 0 zc Critical Value for a Confidence Level, c P(– zc < z < zc ) = c This area = c.

  10. – z.90 0 z.90 Find z0.90 such that 90% of the area under the normal curve lies between z-0.90 and z0.90. P(-z0.90 < z < z0.90 ) = 0.90 .90

  11. – z.90 0 z.90 Find z0.90 such that 90% of the area under the normal curve lies between z-0.90 and z0.90. • P(0< z < z0.90 ) = 0.90/2 = 0.4500 .4500

  12. – z.90 0 z.90 Find z0.90 such that 90% of the area under the normal curve lies between z-0.90 and z0.90. • P( z < z0.90 ) = .5 + 0.4500 = .9500 .9500

  13. Find z0.90 such that 90% of the area under the normal curve lies between z-0.90 and z0.90. • According to Table 5a in Appendix II, 0.9500 lies exactly halfway between two area values in the table (.9495 and .9505). • Averaging the z values associated with these areas gives z0.90 = 1.645.

  14. Common Levels of Confidence and Their Corresponding Critical Values

  15. Confidence Interval for the Mean of Large Samples (n  30)

  16. Confidence Interval for the Mean of Large Samples (n  30)

  17. Confidence Interval for the Mean of Large Samples (n  30)

  18. Confidence Interval for the Mean of Large Samples (n  30)

  19. Confidence Interval for the Mean of Large Samples (n  30)

  20. Confidence Interval for the Mean of Large Samples (n  30)

  21. Create a 95% confidence interval for the mean driving time between Philadelphia and Boston. Assume that the mean driving time of 64 trips was 6.4 hours with a standard deviation of 0.9 hours.

  22. = 6.4 hourss = 0.9 hoursc = 95%, so zc = 1.96n = 64

  23. = 6.4 hourss = 0.9 hoursApproximate  as s = 0.9 hours. 95% Confidence interval will be from

  24. = 6.4 hourss = 0.9 hoursc = 95%, so zc = 1.96n = 64

  25. 95% Confidence Interval: 6.4 – .2205 <  < 6.4 + .2205 6.1795 <  < 6.6205 We are 95% sure that the true time is between 6.18 and 6.62 hours.

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