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Objectives

Objectives. Look at Central Limit Theorem Sampling distribution of the mean. Central Limit Theorem (CLT). Suppose X is random mean  standard deviation not necessarily normal. Terms Concerning Sampling Distributions. Sampling error

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Objectives

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  1. Objectives • Look at Central Limit Theorem • Sampling distribution of the mean

  2. Central Limit Theorem (CLT) • Suppose X is • random • mean  • standard deviation • not necessarily normal

  3. Terms Concerning Sampling Distributions • Sampling error • Sample cannot be fully representative of the population • Variability due to chance – get different values • Standard Error of the mean: • The standard deviation of the sampling distribution of the mean.

  4. CLT (continued) • The mean of several draws from this distribution ( ) is • random • mean of  • standard deviation = • called standard error • approximately normal for large samples • normal for all samples if X is normal

  5. The Central Limit Theorem • For any population of scores, regardless of form, the sampling distribution of the mean will approach a normal distribution as the sample size (N) gets larger. • Furthermore, the sampling distribution of the mean will have a mean equal to µ (population mean), and a standard deviation equal to

  6. Requirements of Central Limit Theorem • Use sample data and the normal curve to reach conclusions about a population • Large, random sample • http://www.ruf.rice.edu/~lane/stat_sim/index.html

  7. What do we mean by random? • Define the population • Identify every member of population • Select from population in such a way that every sample has equal probability of being selected

  8. Biased samples • Non-random selection can result in under-selection or over-selection of subsections of the population. e.g. carry out a internet opinion poll

  9. In summary: sample means • are random • are normally distributed for large sample sizes • distribution has mean  • distribution has standard error • With increase in N • The distribution of means approaches normality • Regardless of parent population’s distribution • The mean of the sampling distribution approaches  • Standard error decreases • Less variability among our sample means

  10. Confidence intervals • Draw a sample, gives us a mean. • is our best guess at µ • For most samples will be close to µ • Point estimate • What if I’d like a range (interval estimate) rather for the possible values of µ? • Use the normal distribution

  11. Confidence interval equation Where = sample mean Z = z value from normal curve based on what confidence level we choose = standard error of the mean

  12. 95% confidence interval • Let’s say we want a 95% confidence interval. • Look up the z-score for p =.025 (since 2.5% above +z, and 2.5% below -z) • p = .025 then z = 1.96* *Recall our key areas under the standard normal distribution curve: + 2 sd (i.e. between a z-score of +2 and -2) encompasses 95% of the area

  13. Confidence interval example • Randomly selected a group of 100 UNT students with a mean score of 40 (s = 6) on some exam. • We guess can we make as to the true mean of UNT students?

  14. 40 + 1.96 • 40 +1.96(.6) • (40 - 1.17) < < (40 + 1.17) • 38.83 < < 41.17

  15. Your turn • Calculate a 99% confidence interval if the mean was 50, s = 10 (n still 100). • 47.43 < µ < 52.57 • What happens to your interval with more variability? Smaller N? Higher percentage?

  16. Important: what a confidence interval means • A 95% confidence interval means that 95% of the confidence intervals calculated on repeated sampling of the same population will contain µ • It does not mean that 95% of the time, the true mean will fall between _ and _ values • Our interval varies with repeated samples, this interval is one of many • http://www.ruf.rice.edu/~lane/stat_sim/conf_interval/index.html

  17. Properties of Confidence Intervals • The wider a confidence interval, the less precise the estimate • The 90% (or lower) confidence interval for an estimate is narrower than the 95% confidence interval • More precise estimate but more chance for error • A 99% confidence interval is wider

  18. Demonstration: grades on a test 80% confidence interval |----------------------------| 85 87.5 90 99% CI |-----------------------------------------------------| 80 87.5 95 Now I’m really confident my interval encompasses the population mean!

  19. Question • How does one know if the confidence interval calculated contains the true population mean?

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