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How are sample means distributed?

How are sample means distributed?. Sampling distribution for a sample mean. Sample means can come from any distribution shape, e.g. skewed, bimodal, etc. But there is a very important theorem that simplifies our work for us…. The Central Limit Theorem .

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How are sample means distributed?

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  1. How are sample means distributed?

  2. Sampling distribution for a sample mean • Sample means can come from any distribution shape, e.g. skewed, bimodal, etc. • But there is a very important theorem that simplifies our work for us…

  3. The Central Limit Theorem • As long as a sample is obtained by random selection and the sampled values are independent of each other, then • As the sample size n increases • The distribution of the sample mean tends toward a Normal distribution with

  4. So, you might think we can go ahead and use the Normal distribution again… Just like we did with proportions with z-tests and z-intervals

  5. But there’s a catch… Today we learn about the t distributions A new distribution closely related to the Normal

  6. Why t ? • By the CLT, the sampling distributions of sample means tends to be Normal as n increases with a mean μ, the same as the population, and a standard deviation of…

  7. Why t ?… HOWEVER, we never really know the population’s standard deviation, so we have to approximate it with the sample standard deviation…

  8. Degrees of freedom • This single step forces us to switch from the Normal to the family of t distributions. • And there is a different t distribution for each sample size. • Each sample size has a different degree of freedom equal to n – 1. • You’ll have to trust me that a degree of freedom is theoretical measure that we can simply use as a label.

  9. Using the t tables or a calculator • Estimate the critical value of t for a 95% confidence interval with 9 degrees of freedom. • Estimate the critical value of t for a 95% confidence interval with 32 degrees of freedom • Estimate the critical value of t for a 95% confidence interval with 400 degrees of freedom. • What do you notice? • Estimate the critical value of t for a 99% confidence interval with 58 degrees of freedom.

  10. Using the t tables or a calculator • Find the P-value for t > 1.372 with 10 degrees of freedom. • Find the P-value for |t| > 2.690 with 45 degrees of freedom. • Find the P-value for t > 1.834 with 30 degrees of freedom. • Find the P-value for |t| > 2.58 with 37 degrees of freedom.

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