1. Thinking... Why isn’t it practical to examine a whole population?

2. Sampling Aims: To look at how the means and standard deviations/variances of samples compare to the original population.

3. Lesson Outcomes Name: To know what the distribution of sample means X is and know what is meant by “unbiased estimators” of sample mean, variance and standard deviation are. Describe: The key features of the distribution of sample means and how they relate to the original population. Explain: How to find an unbiased estimator of the population variance.

4. Maths Ex 5A p122, 5B p126, 5C p131 Next Lesson: More Sampling Aleksandr Mikhailovich Lyapunov 1857-1918: Russian mathematician who proved the central limit theorem we meet today.

5. Sampling It is rarely practical to sample a whole population. Even getting information about all students in college is a challenge. For this reason it is important to know how the key features (mean and standard deviation) of samples compare to the corresponding values of the “parent population”.

6. Averages For years you have been taught that the average of a sample is representative of the data being sampled. You have also been told that to get a more reliable estimate you need _________ Time to learn why.

7. Animation Not sure if this will work. http://onlinestatbook.com/stat_sim /sampling_dist/index.html http://onlinestatbook.com/stat_sim /sampling_dist/index.html We will now look at some key ideas in sampling.

10. Distribution of Means When sampling a population that follows a normal distribution the means of all possible samples (of size n) will create another data set. This dataset also follows a normal distribution. If the original distribution is X~N(μ,σ 2 ) then the means of these samples follow

11. Distribution of Means The distribution of means is So the mean of the sample means is (the population mean). The standard deviation (called the standard error of the sample means) is

12. Using (and hints)

13. None Normal Distributions

15. Central Limit Theorem In very simple terms the central limit theorem states that regardless of the population distribution the sample means will still follow Provided the sample size is large enough (n ≥ 30) though we can usually go ahead whatever.

16. Worth a Mark or Two

17. Unbiased Estimators Because the distribution of sample means has a mean that is the same as the population being sampled then the mean of a sample is already an unbiased estimator. What about variances and standard deviations of samples how do they compare to the variance and standard deviations of the population?

19. Unbiased Estimators of Variance Since the Variance/Standard Deviation of samples is naturally less than that of the population being sampled we amend it to be more representative (often called s 2 /s). This is done by dividing the variance by n-1 rather than n

20. Unbiased Estimators of Variance Note the normal standard deviation can be converted to s by multiplying by

21. Example

22. Round-Up The means of samples of populations with a mean μ and standard deviation σ will follow If the population is already normally distributed then this is always the case. If not normally distributed it will still be the case if n≥30 this is called Central Limit Theorem.

23. Round-Up The mean of samples is, on average, the mean of the population. This means that the mean of a sample is an unbiased estimate for the mean of the population. The variance/standard deviation of a sample is less varied than the population it is from. This is corrected using