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The concept of the sampling distribution of the mean and its relationship with the Central Limit Theorem. It covers the calculation of population mean, sampling distribution of the mean, and standard deviation of the sampling distribution of the mean. The document also discusses the Central Limit Theorem and its significance for inferential statistics.
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Section Q Distribution of the Sample Mean and the Central Limit Theorem Up to this point, the probabilities we have found have been based on individuals in a sample, but suppose we want to find probabilities based on the mean of a sample. In order for us to find these probabilities we need to know determine the sampling distribution of the sample mean. Knowing the sampling distribution of the sample mean will not only allow us to find probabilities, but it is the underlying concept that allows us to estimate the population mean and draw conclusions about the population mean which is what inferential statistics is all about. Sampling Error : The error resulting from using a sample to estimate a population characteristic.
n) is called the sampling distribution of the mean. Note: The larger the sample size the smaller the sampling error tends to be in estimating a population mean,
In other words, the mean of all possible sample means of size n equals the population mean. Example: The following data represent the ages of the winners (age, in years, at time of award given) of the Academy Award for Best Actress for the years 2012 – 2017. 2012: Meryl Streep 62 2013: Jennifer Lawrence 22 2014: Cate Blanchett 44 2015: Julianne Moore 54 2016: Brie Larson 26 2017: Emma Stone 28
b) The following table consisting of all possible samples with size n = 2 and calculate their corresponding means. Sample Mean Sample Mean Sample Mean Sample Mean Sample Mean Sample Mean 62, 62 22, 62 44, 62 54, 62 26, 62 28, 62 62, 22 22, 22 44, 22 54, 22 26, 22 28, 22 62, 44 22, 44 44, 44 54, 44 26, 44 28, 44 62, 54 22, 54 44, 54 54, 54 26, 54 28, 54 62, 26 22, 26 44, 26 54, 26 26, 26 28, 26 62, 28 22, 28 44, 28 54, 28 26, 28 28, 28
(i.e. calculate the mean of the sample means)
population under consideration divided by the square root of n.
σ √n
Example: Using the example above, a) Calculate the population standard deviation, .____________
(i.e. calculate the standard deviation of the sample means) n = _______ ∑ x̅ = ___________ ∑ x̅ 2 = ___________
σ √n^
d) What do you conclude from parts b and c?______________________ Again, the example is an example not a proof.
The Sampling Distribution of the Sample Mean
σ √n
distribution of x.
σ √n
Note: Since the sampling distribution of the sample mean is normally under certain conditions you can use
Examples:
n= _______ μ = ______ σ = ______ b) Find the probability that the mean time spent studying per week is between 8 and 9 hours. c) Find the probability that the mean time spent studying per week is greater than 9.5 hours.
2 ) At an urban hospital the weights of newborn babies are normally distributed, with a mean of 7.2 pounds and standard deviation of 1.2 pounds. Suppose a random sample of 30 is selected. a) What is the sampling distribution of the mean weight of these newborn babies?
n= _______ μ = ______ σ = ______ b) Find the probability the mean weight is less than 6.9 pounds? c) Find the probability the mean weight is between 6.5 and 7.5 pounds? d) Find the probability the mean weight is greater than 8 pounds?
n= _______ μ = ______ σ = ______ b) What is the probability the mean life is less than 38.5? Would this be unusual?