1) Describe sampling distributions and sampling variavility 2) Explain The Central Limit Theorem 3) Explain how confidence intervals are created and what can they tell us about population parameters

FobelloE 2021-03-11 Answered
1) Describe sampling distributions and sampling variavility
2) Explain The Central Limit Theorem
3) Explain how confidence intervals are created and what can they tell us about population parameters
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Expert Answer

Leonard Stokes
Answered 2021-03-12 Author has 98 answers
Step 1
1)
The sampling distributions refers to the probability distributions of all possible values of the sample statistic for example, sample mean.
The sampling variability tells us how an estimate varies between different samples. It is often measured in terms of variance or standard deviation. There are three factors involved in variability,
The size of the population
The size of the sample
The sampling method (with or without replacement)
Step 2
2)
The central limit theorem tells us that in a random sample of size n taken from a population, its sampling distribution can be approximated to a normal distribution by taking a larger sample size. That is, whatever might be the population distribution, the sampling distribution of a sample statistic can be approximated to normal by increasing the sample size.
Step 3
3)
The confidence interval are created using the sample statistic and the margin of error.
CI=x±Zα2(σnn)
=x±ME
Zα2 represents the critical value of the normal distribution and this distribution is used if the population standard deviation is known. For unknown population standard deviation, the sample standard deviation is used with student’s t distribution.
The constructed confidence interval with say 95 or 90 percent confidence level tells us that if repeated samples were to be taken and confidence intervals were to be built, then 95 or 90 percent of these constructed confidence intervals would contain the true value of the parameter (mean).
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Young's modulus is a quantitative measure of stiffness of an elastic material. Suppose that for aluminum alloy sheets of a particular type, its mean value and standard deviation are 70 GPa and 1.6 GPa, respectively (values given in the article ''Influence of Material Properties Variability on Springback and Thinning in Sheet Stamping Processes: A Stochastic Analysis'' (Intl. J. of Advanced Manuf. Tech., 2010: 117–134)). If X¯ is the sample mean Young’s modulus for a random sample of n=16 sheets, where is the sampling distribution of X¯ centered, and what is the standard deviation of the X¯ distribution?