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Unit 3: Inference for Categorical Data: Proportions flashcards

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3.1 Estimators

What is an estimator?
A statistic used to estimate a parameter: a rule such as p̂ = x / n. The point estimate is the single number it gives for your sample. 18 of 75 commuters cycle: p̂ = 0.24 estimates p, the proportion of all commuters who cycle.
What does it mean for an estimator to be unbiased?
The mean of its sampling distribution equals the parameter, so it neither overestimates nor underestimates on average. Individual estimates still miss; unbiased does not mean exact.
Which common estimators are unbiased?
With random samples, p̂ for p, x̄ for μ, and s² (dividing by n − 1) for σ² are unbiased. Dividing by n underestimates σ²; the sample maximum and sample range underestimate the population maximum and range.
How do you justify whether an estimator is biased?
Find or simulate its sampling distribution, find its mean and compare with the parameter. Simulated values averaging 41.9 when the parameter is 50 show it is biased, underestimating. Never judge from one sample.
Does a larger sample remove bias in an estimator?
No. It reduces variability only. An unbiased estimator also gives biased results if the sampling method is biased (convenience, voluntary response, undercoverage).
What notation separates parameters from statistics?
Parameters: p, μ, σ. Sample statistics: p̂, x̄, s. Do not write p or μ for a sample result.

3.2 Sampling Distributions for Sample Proportions

What are the mean and standard deviation of the sampling distribution of p̂?
μp̂ = p, so p̂ is unbiased. σp̂ = √(p(1 − p) / n). With p = 0.7 and n = 84, σp̂ = √(0.21 / 84) = 0.05.
When is the sampling distribution of p̂ approximately normal?
When np ≥ 10 and n(1 − p) ≥ 10, using the given p, not n ≥ 30. With p = 0.7 and n = 84: 58.8 and 25.2, both at least 10.
What is the 10% condition and why is it needed?
When sampling without replacement, n ≤ 0.10N, so observations are close to independent and the σp̂ formula holds. A sample of 100 from 600 members fails: 10% of 600 is 60.
How does sample size affect σp̂?
σp̂ shrinks as n grows: multiply n by 4 to halve it (doubling n divides it by √2). σp̂ = 0.04 at n = 150 becomes 0.02 at n = 600.
How do you find the sample size needed for σp̂ ≤ d?
n ≥ p(1 − p) / d², rounded up.
How do you interpret σp̂ in context?
"In random samples of n individuals, the sample proportion who [success] typically differs from the population proportion p by about σp̂."
How do you find a probability about p̂?
With the normal shape, use z = (p̂ − p) / σp̂ and find the area. A small probability is evidence against a claimed p, not proof it is false.

3.3 Constructing a Confidence Interval for a Population Proportion

What is a confidence interval?
An interval estimate: a range of plausible values for a population parameter, built from one sample, with the shape point estimate ± margin of error.
How do you construct a one-sample z-interval for a population proportion?
p̂ ± z∗ × √[p̂(1 − p̂)/n]. 60 of 150 say yes: p̂ = 0.4, SE = 0.04, MOE = 1.960 × 0.04 = 0.078, 95% interval (0.322, 0.478).
What are the critical values z∗ for common confidence levels?
z∗ = invNorm((1 + C)/2): 1.645 for 90%, 1.960 for 95%, 2.576 for 99%. invNorm(0.90) gives 1.282, the 80% value.
What conditions must be checked for a one-proportion z-interval?
Random: say how the sample was chosen at random, in context. 10%: n ≤ 0.10N when sampling without replacement. Large counts: np̂ ≥ 10 and n(1 − p̂) ≥ 10, using observed successes and failures.
How do you get p̂ and the margin of error from an interval?
For (a, b): p̂ = (a + b)/2 and MOE = (b − a)/2. (0.55, 0.63) gives p̂ = 0.59 and MOE = 0.04. The width is 2 × MOE.
How do you find the sample size for a chosen margin of error?
n = (z∗/MOE)² × p̂(1 − p̂), rounded up, using p̂ = 0.5 if there is no estimate. 95% with MOE 0.04: (1.960/0.04)² × 0.25 = 600.25, so 601.
How should the parameter be defined for a proportion interval?
With three parts: the proportion, the response (what counts as a success) and the population, for example "the proportion of all adults in the town who say yes". It is not a proportion of the sample.

3.4 Justifying a Claim Based on a Confidence Interval for a Population Proportion

How do you interpret a confidence interval for p?
"We are C% confident that the interval from a to b captures the proportion of all [population] who [response]." Not "a 95% probability that p is between a and b", and not where 95% of individuals' values lie.
How do you interpret the confidence level?
If we took many random samples of size n and built a C% interval from each, about C% of the intervals would capture the true proportion. Of 100 such 95% intervals, about 95 capture p.
How do you use a confidence interval to judge a claimed value?
Outside the interval: convincing evidence against it. Inside: plausible, with no convincing evidence against it, but not proved. For (0.62, 0.70), 0.75 is rejected and 0.65 is plausible.
How does the confidence level affect the interval?
With the same data, a higher level raises z∗, so the MOE and width grow; the SE is unchanged. Higher confidence means less precision.
How does sample size affect a confidence interval?
Larger n lowers the SE, MOE and width. Multiplying n by k divides the width by about √k: 4 × n halves it, 9 × n divides it by about 3.
What does the margin of error not account for?
Bias. It covers random sampling variation only, and the interval means something only if the random, 10% and large-counts conditions hold.
How do you use an interval to support a one-sided claim?
If the whole interval lies on one side of a boundary (for example all above 0.5), that is convincing evidence for the claim on that side. If it straddles the boundary, the evidence is not convincing either way.

3.5 Setting Up a Test for a Population Proportion

How do you write hypotheses for a one-proportion test?
H₀: p = p₀ (status quo, contains "="). Hₐ: p > p₀, p less than p₀ or p ≠ p₀ (never "="), from the question's wording, not the sample result. Define p as the true proportion of the population with the response.
Which hypothesis holds the researcher's claim?
Hₐ, the researcher's suspicion. "At least" or "at most" claims are tested at the boundary, H₀: p = p₀.
What is the normality condition for a one-proportion z-test?
Expected counts under H₀: np₀ ≥ 10 and n(1 − p₀) ≥ 10, not observed counts. H₀: p = 0.10 with n = 60 fails, since np₀ = 6.
What conditions does a one-proportion z-test need?
Random sample from the population; n ≤ 10% of N when sampling without replacement; np₀ ≥ 10 and n(1 − p₀) ≥ 10. 400 from 3,000 fails the 10% condition (limit 300).
Why is the null distribution of p̂ centred at p₀?
The test assumes H₀ is true. That is why tests use p₀ in the normality check and standard error, while intervals use p̂.
Give hypotheses for: a café says 30% order oat milk; the owner thinks it has changed.
H₀: p = 0.30; Hₐ: p ≠ 0.30, where p is the true proportion of the café's customers who order oat milk.

3.6 p-Values

What is a p-value?
The probability, assuming H₀ is true, of a test statistic as extreme as the observed one or more extreme, in the direction of Hₐ. It is not the probability that H₀ is true.
How do you interpret a small or large p-value?
Small: the result would be unusual if H₀ were true, so it is evidence for Hₐ; smaller is more convincing. Not small: no convincing evidence for Hₐ, but this is not evidence that H₀ is true.
How do you find a p-value from z for each alternative?
Hₐ: p > p₀ uses P(Z ≥ z). Hₐ: p less than p₀ uses P(Z ≤ z). Hₐ: p ≠ p₀ uses 2P(Z ≥ |z|). Hₐ: p less than p₀ with z = −0.85 gives about 0.1977; two-sided z = 2.05 gives about 0.0404.
How do you estimate a p-value from a simulation?
Find the proportion of simulated values under H₀ at least as extreme as the observed one, in the direction of Hₐ, including the observed value. 9 of 250 at or above gives 0.036.
How do you write a p-value interpretation?
"Assuming the true proportion of [population] who [response] is p₀, there is a [p-value] probability of getting a sample proportion of p̂ or more extreme in a random sample of n, by chance alone."
Is 1 − p-value the probability that Hₐ is true?
No. The p-value is a probability about the data assuming H₀, not about which hypothesis is true.

3.7 Carrying Out a Test for a Population Proportion

What are the four parts of a significance test write-up?
Hypotheses (with p defined in context and α), method and conditions, calculations (test statistic and p-value), and a conclusion in context.
What is the test statistic for a one-proportion z-test?
z = (p̂ − p₀) / √[p₀(1 − p₀) / n], using p₀, not p̂, in the standard error. n = 80, x = 36, H₀: p = 0.35: p̂ = 0.45, z ≈ 1.88, one-sided p-value ≈ 0.0304.
What is the significance level α?
Chosen before collecting data, it is the probability of rejecting H₀ when H₀ is true. Use 0.05 if none is given and say so; never change it after seeing the p-value.
How do you make the decision in a test?
p-value ≤ α: reject H₀, convincing evidence for Hₐ. p-value above α: fail to reject H₀, not convincing evidence for Hₐ. Never "accept H₀" or "prove".
How do you write a test conclusion?
"Because the p-value of 0.0304 is greater than α = 0.01, we fail to reject H₀. There is not convincing statistical evidence that the true proportion is greater than 0.35." Link p-value and α, and conclude about p, not p̂.
What does the test statistic z measure?
How many standard errors p̂ is from p₀, assuming H₀ is true. Under H₀ with the conditions met, z is approximately standard normal (the null distribution).

3.8 Potential Errors When Performing Tests

What are Type I and Type II errors?
Type I: H₀ is rejected (convincing evidence for Hₐ) but H₀ is true. Type II: H₀ is not rejected but Hₐ is true. Describe both the conclusion and the true situation in context.
What is the power of a test?
The probability of correctly rejecting H₀ when it is false, for a stated alternative value. P(Type II error) = β = 1 − power; with power 0.83, β = 0.17. A good design target is power ≥ 0.80.
What is the probability of a Type I error?
α, the significance level chosen before collecting data. P(Type II) is 1 − power, not 1 − α.
How can you increase the power of a test?
Larger n, a smaller standard error, a true value further from p₀, or a larger α (which raises Type I risk). A larger sample raises power; it does not lower α.
Which error is possible after a given decision?
Reject H₀: only a Type I error is possible. Fail to reject: only a Type II error. A test rejecting H₀ with p-value 0.02 could only have made a Type I error.
How do you choose α based on the consequences of errors?
If a Type I error is more serious, choose a small α. If a Type II error is more serious, choose a larger α and/or a larger n. A smaller α is not always safer: it raises Type II risk.
Describe a Type II error for H₀: p = 0.20, Hₐ: p less than 0.20 (takeaway orders).
The test does not find convincing evidence that fewer than 20% of the café's orders are takeaway, when really fewer than 20% are.

3.9 Sampling Distributions for the Difference Between Sample Proportions

What are the mean and standard deviation of p̂1 − p̂2?
Mean: p1 − p2. Standard deviation: √[p1(1 − p1)/n1 + p2(1 − p2)/n2]. With p1 = 0.5, n1 = 100, p2 = 0.4, n2 = 150: mean 0.1, SD = √(0.0025 + 0.0016) ≈ 0.0640.
Why do you add variances, not standard deviations, for a difference?
Variability from two independent samples combines, so the variances add (even though the proportions are subtracted). The SD of p̂1 − p̂2 is larger than either SD alone but smaller than their sum.
What conditions apply to the sampling distribution of p̂1 − p̂2?
Randomization: two independent random samples or random assignment. 10%: n1 ≤ 0.10N1 and n2 ≤ 0.10N2 when sampling (not for an experiment). Large counts: n1p1, n1(1 − p1), n2p2, n2(1 − p2) all at least 10.
What happens to the SD of p̂1 − p̂2 if both sample sizes are multiplied by 4?
It is divided by √4 = 2, so it halves.
Why must the order of subtraction be stated?
Swapping the order changes the sign of the mean (the SD is unchanged), so without a stated order such as "town A − town B" the sign has no meaning.
How do you find a z-score for an observed difference in sample proportions?
z = [(p̂1 − p̂2) − (p1 − p2)] / σ(p̂1 − p̂2), with an approximately normal shape when all four large counts are met. A small probability gives reason to doubt a claim, not proof.

3.10 Constructing a Confidence Interval for the Difference Between Two Population Proportions

How do you construct a two-sample z-interval for p1 − p2?
(p̂1 − p̂2) ± z∗ × √[p̂1(1 − p̂1)/n1 + p̂2(1 − p̂2)/n2]. 60 of 100 and 45 of 100: estimate 0.15, SE ≈ 0.0698, MOE ≈ 0.137, 95% interval (0.013, 0.287).
Do you pool proportions for a confidence interval for p1 − p2?
No. Pooling is for tests, which assume p1 = p2. The interval uses each sample's own p̂.
What is the normality condition for a two-proportion interval?
The four observed counts n1p̂1, n1(1 − p̂1), n2p̂2 and n2(1 − p̂2) must all be at least 10. A group with 9 failures fails the condition.
How do you recover the estimate and margin of error from an interval for p1 − p2?
Point estimate = (a + b)/2; MOE = (b − a)/2; SE = MOE ÷ z∗. (0.04, 0.16) gives 0.10 and 0.06.
What happens to an interval for p1 − p2 if the order is reversed?
(a, b) becomes (−b, −a); the width is unchanged.
How should the parameter be defined for a two-proportion interval?
As the difference in true proportions (order stated), with the response and both populations or treatments. Do not define it with the samples.

3.11 Justifying a Claim Based on a Confidence Interval for the Difference Between Two Population Proportions

How do you interpret a confidence interval for p₁ − p₂?
"We are C% confident that the interval from a to b captures the difference (population 1 − population 2) in the proportions of all [individuals] who [response]." Use percentage points in words.
What does an interval for p₁ − p₂ that contains 0 tell you?
0 is plausible, so there is not convincing evidence of a difference. It does not prove the proportions are equal. (−0.05, 0.02) is an example.
What does an interval for p₁ − p₂ that is all positive tell you?
Convincing evidence of a difference, with p₁ above p₂. (0.03, 0.11) gives that conclusion; all negative values would show p₁ below p₂.
How do you judge a claimed difference with an interval?
Inside: plausible, no convincing evidence against it. Outside: convincing evidence against it. With (0.03, 0.11), a claimed difference of 8 percentage points (0.08) is plausible.
How do you interpret the confidence level for a difference in proportions?
In repeated random sampling with the same sample sizes from the same populations, about C% of intervals built this way would capture the true difference p₁ − p₂. The 95% is about the method.
When can a two-proportion interval support cause and effect?
Only with random assignment in an experiment. Independent random samples allow generalising to the populations, not a causal conclusion.

3.12 Setting Up a Test for the Difference Between Two Population Proportions

What hypotheses are used for a two-sample z-test for proportions?
H₀: p₁ = p₂ (or p₁ − p₂ = 0). Hₐ: p₁ > p₂, p₁ less than p₂ or p₁ ≠ p₂, chosen from the question's wording before seeing data. "Change" or "affects" has no direction, so Hₐ: p₁ ≠ p₂.
How do you calculate the pooled proportion?
p̂c = (x₁ + x₂)/(n₁ + n₂), pooling the counts, not averaging p̂₁ and p̂₂. 30 of 100 and 60 of 150 give p̂c = 90/250 = 0.36.
What is the normality condition for a two-proportion test?
n₁p̂c, n₁(1 − p̂c), n₂p̂c and n₂(1 − p̂c) all at least 10. For p̂c = 0.36 with n = 100 and 150: 36, 64, 54, 96. Not observed counts, not n ≥ 30.
When is the 10% condition not needed for a two-proportion test?
In a randomized experiment, such as 200 volunteers randomly assigned to two diets. For independent random samples, check n₁ ≤ 0.10N₁ and n₂ ≤ 0.10N₂.
When can a two-sample z-test for proportions not be used?
For paired or before-and-after data, which are not independent samples. Independent samples have no individual in both and no pairing.
How are the parameters defined in an experiment?
The "populations" are the treatments: p₁ is the true proportion of subjects like these who would respond with treatment 1, and p₂ likewise for treatment 2.

3.13 Carrying Out a Test for the Difference Between Two Population Proportions

What is the test statistic for a two-sample z-test for proportions?
z = (p̂₁ − p̂₂ − 0) / √[p̂c(1 − p̂c)(1/n₁ + 1/n₂)], using the pooled proportion. The unpooled standard error belongs to the confidence interval.
Why are the samples pooled in a two-proportion test?
The test is calculated assuming H₀: p₁ = p₂ is true, so both samples estimate one common proportion. 30 of 80 and 54 of 120 give p̂c = 84 ÷ 200 = 0.42, not the average 0.4125.
How do you find the p-value in a two-proportion z-test?
Hₐ: p₁ > p₂ uses P(Z ≥ z); p₁ less than p₂ uses P(Z ≤ z); p₁ ≠ p₂ uses 2 × P(Z ≥ |z|). z = 1.80 gives 0.0359 one-sided, 0.0719 two-sided.
How do you interpret the p-value of a two-proportion test?
The probability, assuming the true proportions are equal, of a difference in sample proportions at least as extreme as the one observed, in the direction of Hₐ.
How do you conclude a two-proportion test?
State the p-value and α with the decision: 0.0359 is greater than α = 0.01, so fail to reject H₀; not convincing evidence for Hₐ. Conclude about the true proportions, never "accept H₀".
What scope of conclusion does a two-proportion test allow?
Independent random samples support conclusions about the populations; random assignment supports cause and effect for the treatments.

3.14 Setting Up a Chi-Square Test for Homogeneity or Independence

How do you choose between chi-square tests for homogeneity and independence?
By the design, not the table's shape. Homogeneity: independent random samples from 2+ populations (or 2+ treatments), one variable. Independence: one random sample, two variables per individual.
What are the hypotheses for a chi-square test for homogeneity?
H₀: no difference in the distribution of [variable] across [populations or treatments]. Hₐ: there is a difference, not "all distributions are different".
What are the hypotheses for a chi-square test for independence?
H₀: no association between [variable 1] and [variable 2] in [population]. Hₐ: there is an association. An association in an observational study is not cause.
What are the properties of chi-square distributions?
They take only positive values and are skewed right. As the degrees of freedom increase, the skew weakens and the peak moves right, so df = 2 is more skewed than df = 12.
What conditions apply to chi-square tests for two-way tables?
Randomization (independent random samples or random assignment, or one random sample); 10% condition when sampling (not for an experiment); all expected counts greater than 5, not observed counts.
What does the chi-square statistic measure?
The distance between observed and expected counts, relative to the expected counts. It is never negative, larger values are stronger evidence against H₀, and the p-value is always a right-tail area.
What can you do if the expected counts condition fails?
Do not carry out the test as it stands: take a larger sample, or combine categories if that still answers a sensible question.

3.15 Carrying Out a Chi-Square Test for Homogeneity or Independence

How do you calculate an expected count in a two-way table?
(row total × column total) ÷ table total. Row 60, column 45, table 150: 60 × 45 ÷ 150 = 18. Keep decimals; do not round to whole numbers.
How do you calculate the chi-square statistic?
Sum of (observed − expected)² ÷ expected over every cell, using counts, not percentages, and leaving out totals. Observed 12 and expected 20 give a component of 3.2.
What are the degrees of freedom for a chi-square test on a two-way table?
(rows − 1)(columns − 1), counting category rows and columns only. 3 rows and 4 columns: df = 6. Not r × c or (r × c) − 1.
How do you find the p-value of a chi-square test?
P(χ² ≥ observed value) for that df: always the right tail, never doubled. A table gives a range; technology gives a value.
How do the homogeneity and independence tests differ in calculation?
They do not. The calculations are identical; only the hypotheses, the randomization condition and the conclusion wording differ.
How do you write a conclusion for a chi-square test?
Homogeneity: "Because the p-value is less than α, we reject H₀; there is convincing evidence that the distribution of [variable] differs across [populations]." Independence: about an association in the population. Never "proves independent".

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