Math 183 - Part 5 (Statistical Inference)

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30 Terms

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Statistical inference
Using sample data to make conclusions about a population parameter.
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Hypothesis testing
A formal method for deciding whether observed data provide evidence against a null hypothesis.
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Null hypothesis (H₀)
The default claim or status quo, such as “the roulette wheel is fair” (e.g., p = p₀).
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Alternative hypothesis (H₁)
The claim we look for evidence for, such as “the roulette wheel is biased” (e.g., p > p₀ or p ≠ p₀).
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Test statistic
A function of the data used to assess how unusual the observed result is under the null hypothesis.
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Significance level (α)
The maximum probability of making a Type I error (commonly set at 0.05 or 0.01).
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Type I error
Rejecting the null hypothesis when it is actually true (false positive).
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Type II error
Failing to reject the null hypothesis when the alternative is true (false negative).
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Critical value
A threshold set so that the chance of a Type I error is at most α.
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P-value
The probability of seeing data as extreme as (or more extreme than) the observed data, if the null hypothesis is true.
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Decision rule
If the p-value ≤ α, reject the null hypothesis; if the p-value > α, do not reject the null.
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One-sided test
A test where the alternative hypothesis is directional (e.g., p > p₀).
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Two-sided test
A test where the alternative is non-directional (e.g., p ≠ p₀); rejection region is in both tails.
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Exact binomial test
A hypothesis test that uses the binomial distribution to compute probabilities exactly, without approximation.
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Normal approximation
For large n, binomial probabilities are approximated using the normal distribution with the same mean and variance.
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Z-score
A standardized test statistic: z = (observed − expected) / standard deviation under H₀.
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Confidence interval (CI)
A range of values that, with a certain confidence level, is likely to contain the true population parameter.
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Point estimate
A single value (such as the sample proportion) that serves as a best guess for a population parameter.
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Rule of thumb for normal approximation
Normal approximation is appropriate if n×p̂ ≥ 10 and n×(1−p̂) ≥ 10.
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Relationship of CI and tests
A value of the parameter is in the CI if and only if it would not be rejected by a two-sided test at significance level α.
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p-value and evidence
The smaller the p-value, the stronger the evidence against the null hypothesis.
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Step 1 in inference
State the parameter of interest and the hypotheses (H₀ and H₁).
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Step 2 in inference
Collect the data and compute the sample statistic (e.g., sample proportion).
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Step 3 in inference
Calculate the p-value or construct a confidence interval.
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Step 4 in inference
Make a decision (reject or do not reject H₀) and interpret in context.
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Interpretation of CI
A 95% CI means that in 95% of similarly-constructed intervals from repeated samples, the interval will contain the true parameter.
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Alpha and errors
α is the chance of Type I error; reducing α reduces false positives but may increase false negatives.
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Sample size and inference
Larger samples lead to smaller standard errors, narrower confidence intervals, and more powerful tests.
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Model assumptions
Inference is valid only if the probability model and assumptions (e.g., independence, correct distribution) are appropriate.