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Review

Review. Confidence Intervals Sample Size. Estimator and Point Estimate. An estimator is a “sample statistic” (such as the sample mean, or sample standard deviation) used to approximate a population parameter.

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Review

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  1. Review • Confidence Intervals • Sample Size

  2. Estimator and Point Estimate An estimator is a “sample statistic” (such as the sample mean, or sample standard deviation) used to approximate a population parameter. A Point Estimate is a single value or point used to approximate a population parameter. A point estimator may be biased or unbiased.

  3. Central Limit Theorem Take ANY random variable X and compute m and s for this variable. If samples of size n are randomly selected from the population, then: 1) For large n, the distribution of the sample means, will be approximately a normal distribution, 2) The mean of the sample means will be the population mean m and 3) The standard deviation of the sample means will be

  4. Confidence Intervals The Confidence Interval is expressed as: E is called the margin of error. For samples of size > 30,

  5. Sample Size The sample size needed to estimate m so as to be (1-a)*100 % confident that the sample mean does not differ from m more than E is: …round up

  6. Practice Problems • #7.11 page 329 • #7.19 page 331 • #7.21 page 331

  7. Small Samples What happens if and n is small (n < 30)? Our formulas from the last section no longer apply.

  8. Small Samples What happens if and n is small (n < 30)? Our formulas from the last section no longer apply. There are two main issues that arise for small samples: 1) s no longer can be approximated by s 2) The CLT no longer holds. That is the distribution of the sampling means is not necessarily normal.

  9. t- distribution If we have a small sample (n < 30) and wish to construct a confidence interval for the mean we can use a t-statistic, provided the sample is drawn from a normally distributed population.

  10. t-distributions s is unknown so we use s (the sample standard deviation) as a point estimate of s. We convert the nonstandard t-distributed problem to a standard t-distributed problem through the use of the standard t-score

  11. t-distributions • Mean 0 • Symmetric and bell-shaped • Shape depends upon the degrees of freedom, which is one less than the sample size. df = n-1 • Lower in center, higher tails than normal. • See Table inside front cover in text

  12. Example In n=15 and after some calculation a/2=0.025, t0.025 = 2.145

  13. Confidence Interval for the mean when s is unknown and n is small The (1- a)*100% confidence interval for the population mean m is The margin of error E, is in this case N.B. The sample is assumed to be drawn from a normal population.

  14. Confidence Intervals for a small sample population mean The Confidence Interval is expressed as: The degrees of freedom is n-1.

  15. Example The following are the heat producing capabilities of coal from a particular mine (in millions of calories per ton)8,500 8,330 8,480 7,960 8,030Construct a 99% confidence interval for the true mean heat capacity. Solution:sample mean is 8260.0 sample Std. Dev. is 251.9 degrees of freedom = 4 a = 0.01 7741.4  m 8778.6

  16. Confidence intervals for a population proportion The objective of many surveys is to determine the proportion, p, of the population that possess a particular attribute. If the size of the population is N, and X people have this attribute, then as we already know, is the population proportion.

  17. Confidence intervals for a population proportion If the size of the population is N, and X people have this attribute, then as we already know, is the population proportion. The idea here is to take a sample of size n, and count how many items in the sample have this attribute, call it x. Calculate the sample proportion, . We would like to use the sample proportion as an estimate for the population proportion.

  18. Therefore at the (1-a)*100 % level of confidence, the Error estimate of the population proportion is At the (1-a)*100 % level of confidence, the confidence interval for the population proportion is :

  19. Determining Sample Size In calculating the confidence interval for the population proportion we used Perhaps we might be interested in knowing how large a sample we should use if we are willing to accept a margin of error E with a degree of confidence of 1-a.

  20. Determining Sample Size If we already have an idea of the proportion (either through a pilot study, or previous results) one can use If we have no idea of what the proportion is then we use

  21. In Class Exercises • #7.31, 7.36, 7.41 Pages 341, 342 • #7.50, 7.57 on page 349, 350 • #7.75, 7.78 Pages 356, 357

  22. Shortcut for finding za/2 • Recall that as n  the Student’s T-distribution approaches the normal distribution. • Look at T-table inside front cover, the last row represents the values of tn-1, a/2, as n becomes large which is essentially za/2. • Therefore, for some common values of a we are able to find za/2 quite quickly. • z0.025 =1.960, z0.10 =1.282

  23. Homework • Review Chapter 7.3-7.5 • Read Chapters 8.1-8.3 • Quiz on Tuesday: Chapter 5

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