BEES2041 Lecture Notes - Lecture 2: Central Limit Theorem, Null Hypothesis, Confidence Interval

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BEES2041 Lecture 2
Some Simple tests
One sample tests
-What does our sample tell us about population mean?
-what is likelihood of sample coming from a given population?
Two sample tests
-are two samples different?
Confidence interval for the population mean
-remember sample (known) is an estimate of population mean (unknown)
-can use sample mean to give a likely range for population mean
-Probability distribution of sample means from a given population is known (t)
-Conversely, samples can then be used to infer a likely range for population mean
~Means sampled from normal distribution, are normally distributed (Central limit
theorem)
95% of sample means fall within 1.96 standard errors of population mean
-Unfortunately, z scores cant be used very often as we rarely know variance of population
(only estimates from samples)
-Students t distribution is standardised distribution of sample means taken from normally
distributed populations
-t has a symmetrical distribution
-varies with sample size
-very similar to normal when n> 30
-95% confidence interval for population mean becomes:
~Distribution of sample means
95% of samples from a population with mean will be within this region
Hypothesis tests for a single population
-T distribution can also be used to test hypotheses about a single population
-e.g. is the population mean equal to a specific value
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Document Summary

Remember sample (known) is an estimate of population mean (unknown) Can use sample mean to give a likely range for population mean. Probability distribution of sample means from a given population is known (t) Conversely, samples can then be used to infer a likely range for population mean. ~means sampled from normal distribution, are normally distributed (central limit theorem) 95% of sample means fall within 1. 96 standard errors of population mean. Unfortunately, z scores cant be used very often as we rarely know variance of population (only estimates from samples) Students t distribution is standardised distribution of sample means taken from normally distributed populations. 95% of samples from a population with mean will be within this region. T distribution can also be used to test hypotheses about a single population. E. g. is the population mean equal to a specific value. Does your sample come from a population with known properties. Does a sample come from a known population.

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