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Sampling Distribution. Vocabulary. ________ – the entire collection of individuals ________ a subset of population (used in the study) _________ – a number that describes the population _________ – a quantity computed from a sample
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Vocabulary • ________– the entire collection of individuals • ________ a subset of population (used in the study) • _________ – a number that describes the population • _________ – a quantity computed from a sample • __________– the distribution of values taken by the statistic in all possible samples of the same size from the same population
Describing Sampling Distribution 1. center 2. Spread 3. Shape
General Properties of Sampling Distributions of x Rule 1: Rule 2: This rule is approximately correct as long as no more than 10% of the population is included in the sample.
Example Consider a population that consists of the numbers 1, 2, 3, 4 and 5 generated in a manner that the probability of each of those values is 0.2 no matter what the previous selections were. This population could be described as the outcome associated with a spinner such as given below. The distribution is next to it.
If the sampling distribution for the means of samples of size two is analyzed, it looks like
Sampling distribution n = 2 1 1 2 2 3 3 4 4 5 5 The sampling distribution of means of samples with n=2 The original distribution
1 2 3 4 5 1 2 3 4 5 Sampling distribution n = 3 Sampling distribution n = 4 Sampling distributions for n=3 and n=4 were calculated and are illustrated below.
2 3 4 Means (n=30) 2 3 4 Means (n=60) 2 4 3 Means (n=120) Simulations To illustrate the general behavior of samples of fixed size n, 10000 samples each of size 30, 60 and 120 were generated from a uniform distribution and the means calculated. Probability histograms were created for each of these (simulated) sampling distributions. Notice all three of these look to be essentially normally distributed. Further, note that the variability decreases as the sample size increases.
Rule 3: When the population distribution is normal, the sampling distribution of x is alsonormal for any sample size n.
How about if the population distribution is NOT normal?
The paper “Is the Overtime Period in an NHL Game Long Enough?” (American Statistician, 2008) gave data on the time (in minutes) from the start of the game to the first goal scored for the 281 regular season games from the 2005-2006 season that went into overtime. The density histogram for the data is shown below. Let’s consider these 281 values as a population. The distribution is strongly positively skewedwith meanm = 13 minutes and with a median of 10 minutes. Using Minitab, we will generate samples of the following sample sizes from this distribution: n = 5, n = 10, n = 20, n = 30.
What do you notice about the standard deviations of these histograms? Are these histograms centered at approximately m = 13? What do you notice about the shape of these histograms? These are the density histograms for the samples
Rule 4:Central Limit Theorem (CLT) When n is sufficiently large, the sampling distribution of x is well approximated by a normal curve, even when the population distribution isnotitself normal. How large is “sufficiently large” anyway? CLT can safely be applied if n exceeds 30.
Skewed distribution Simulations To further illustrate the general behavior of samples of fixed size n, 10000 samples each of size 4, 16 and 32 were generated from the positively skewed distribution pictured below. Notice that these sampling distributions all all skewed, but as n increased the sampling distributions became more symmetric and eventually appeared to be almost normally distributed.
Objectives Review • How do you describe a sampling distribution? • CLT?????
Class work (F) I, II and III
