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

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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  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.

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

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

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.h tml http://www.ruf.rice.edu/~lane/stat_sim/index.h tml

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

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

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

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

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

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

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?

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

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?

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

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

Demonstration: grades on a test 80% confidence interval |----------------------------| 85 87.5 90 99% CI |-----------------------------------------------------| 8087.5 95 Now I’m really confident my interval encompasses the population mean!

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

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