AP Stats Notes – Central Limit Theorem page 3
When population parameters are not known, sample statistics are used to estimate them.
Sampling variability – values of sample statistics vary from sample to sample
Sampling distribution – the pattern displayed by the values of a sample statistic for all possible samples of the same size from the same population.
· Population parameters are fixed, but samples vary
· In the long run, patterns emerge in the variations
· The sampling distribution is the patterns that will theoretically be observed when all possible samples of the same size are taken from the same population.
Central Limit Theorem for Sample Proportions – describes the patterns of the sampling distribution in terms of …
· Shape – normal distribution, IFF and .
· Center - ( is the mean of all the sample proportions)
· Spread -
Central Limit Theorem for Sample Means – describes the patterns of the sampling distribution in terms of …
· Shape – normal distribution, IFF or the population being sampled is normally distributed.
· Center - ( is the mean of all the sample means)
· Spread -
Statistical significance – if a sample result is unlikely to occur simply due to sampling variability alone. If a result not likely to occur by chance alone, it is described as statistically significant.
Reese’s Pieces Experiment
The proportion of orange candies was = 0.45. We took samples of size 25, so n = 25.
What does the CLT say about the sampling distribution for samples of size 25 from this population?
· Shape – normal distribution, IFF and . Are these conditions met?
Yes.
The sampling distribution will be normal.
· Center - = 0.45
· Spread - = = 0.0995
By the empirical rule, 95% of sample proportions should be between (roughly) 0.25 and 0.65. We should be surprised to get a sample proportion that is not between these values. If we did, that result would be considered statistically significant. It might lead us to conclude that the true proportion of orange candies is not really 0.45.