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Introduction to Inference
- Estimates of sample mean for samples are always relatively close to the populationparameter μ.
- 9 5% of all sample means will be within roughly 2 standard deviations (2*s/√n) of the population parameter μ.
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A level C confidence interval for a parameter has two parts:
- (1) An interval calculated from the data, usually of the form
- estimate ± margin of error
(2) A confidence level C, which gives the probability that the interval will capture the true parameter value in repeated samples, or the success rate for the method.
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confidence interval
for a population mean can be expressed from a sample mean as:
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confidence level C
(in %) indicates the success rate of the method that produces the interval. It represents the area under the normal curve within ± m of the center of the curve.
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(in %)
confidence level C
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Standardizing the normal curve using z
Ex. For a 98% confidence level, z*=2.326
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Hypotheses tests
- 1. test of statistical significance
- 2. null hypothesis
- 3. alternative hypothesis
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test of statistical significance
- tests a specific hypothesis using sample data to decide on the
- validity of the hypothesis.
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hypothesis
is an assumption, or a theory about the characteristics of one or more variables in one or more populations
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null hypothesis
is the statement being tested. It is a statement of “no effect” or “no difference,” and it is labeled H0.
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alternative hypothesis
is the claim we are trying to find evidence for, and it is labeled Ha.
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two-tail or two-sided test
- of the population mean has these null and alternative hypotheses:
- H0: μ = [a specific number] Ha: μ Å [a specific number]
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one-tail or one-sided test
- of a population mean has these null and alternative hypotheses:
- H0: μ = [a specific number] Ha: μ < [a specific number] OR
- H0: μ = [a specific number] Ha: μ > [a specific number]
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Tests for a population mean
To test the hypothesis H0: μ = μ0 based on an SRS of size n from a Normal population with unknown mean μ and known standard deviation σ, we rely on the properties of the sampling distribution N(μ, σ√n).
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p-value
is the area under the sampling distribution for values at least as extreme, in the direction of Ha, as that of our random sample.
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The P-value
- Tests of statistical significance quantify the chance of obtaining a particular random sample result if the null hypothesis were true. This quantity is the P-value.
- This is a way of assessing the “believability” of the null hypothesis given the evidence provided by a random sample.
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P-value in one-sided and two-sided tests
To calculate the P-value for a two-sided test, use the symmetry of the normal curve. Find the P-value for a one-sided test and double it.
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Interpreting a P-value
Could random variation alone account for the difference between the null hypothesis and observations from a random sample?
- A small P-value implies that random variation because of the sampling process alone is not likely to account for the observed difference.
- With a small P-value, we reject H0. The true property of the population is significantly different from what was stated in H0.
Thus small P-values are strong evidence AGAINST H0.
Oftentimes, a P-value of 0.05 or less is considered significant: The phenomenon observed is unlikely to be entirely due to chance event from the random sampling.
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The significance level a
- The significance level, α, is the largest P-value tolerated for rejecting a true null hypothesis (how much evidence against H0 we require). This value is decided arbitrarily before
- conducting the test.
If the P-value is equal to or less than α (p ≤ α), then we reject Ho. If the P-value is greater than α (p > α), then we fail to reject Ho.
When the z score falls within the rejection region (shaded area on the tail-side), the P-value is smaller than α and you have shown statistical significance.
- Rejection region for a two-tail test of μ with α = 0.05 (5%)
- A two-sided test means that α is spread between both tails of the curve, thus:
- a middle area C of 1 − α = 95%, and
- an upper tail area of α /2 = 0.025
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Confidence intervals to test hypotheses
- Because a two-sided test is symmetrical, you can also use a confidence interval to test a twosided
- hypothesis.
- In a two-sided test, C = 1 – α.
- C confidence level
- α significance level
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Logic of confidence interval test
- A confidence interval gives a black and white answer: Reject or don’t reject H0. But it also estimates a range of likely values for the true population mean μ.
- A P-value quantifies how strong the evidence is against the H0. But if you reject Ho, it doesn’t provide any information about the true population mean μ.
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