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Unit 4: Inference for Quantitative Data: Means flashcards

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4.1 Sampling Distributions for Sample Means

What are the mean and standard deviation of the sampling distribution of x̄?
μx̄ = μ, so x̄ is unbiased for μ and the mean does not depend on n. σx̄ = σ/√n, which needs independent values. σ = 30 with n = 25 gives σx̄ = 6.
When is the sampling distribution of x̄ normal or approximately normal?
Normal if the population is normal (any n); approximately normal by the central limit theorem if n ≥ 30. An extremely skewed population may need n well above 30; a non-normal population with n below 30 has no normal model.
How does sample size affect σx̄?
Square-root law: n × 4 halves σx̄ and n × 9 divides it by 3. To halve σx̄ from a sample of 20, use 80. To reach σx̄ = k, n ≥ (σ/k)², rounded up.
How do you find a probability about a sample mean?
z = (x̄ − μ)/(σ/√n), not (x̄ − μ)/σ. Normal population, μ = 50, σ = 6, n = 16: σx̄ = 1.5, P(x̄ above 53) has z = 2, about 0.0228.
What conditions apply to the sampling distribution of x̄?
Randomization (random sample, for independence and an unbiased x̄); 10% (n ≤ 10% of N when sampling without replacement, so σ/√n is accurate); normal shape (normal population or n ≥ 30).
What does σx̄ describe?
The typical distance of a sample mean from μ, not the spread of individual values. Sample means vary less than individuals.
Does the central limit theorem make sample data normal?
No. It is about the distribution of x̄ over many samples, not the data in one sample.

4.2 Constructing a Confidence Interval for a Population Mean or Population Mean Difference

Why use a t-distribution for a confidence interval for a mean?
σ is rarely known, so the standard error s/√n is used, and standardising with s gives a t statistic with df = n − 1. Using z∗ when σ is unknown is a mistake.
How do t-distributions compare with the standard normal?
Symmetric, bell-shaped and centred at 0, but with heavier tails. As df grows they approach the standard normal. Heavier tails make t∗ larger than z∗, so intervals are wider, especially for small n.
How do you construct a one-sample t-interval for a mean?
x̄ ± t∗ × s/√n, with df = n − 1. For a sample of 36 with s = 12 cm, SE = 2 cm. For 90% with n = 20, df = 19 and t∗ ≈ 1.729.
How do you analyse matched-pairs data?
Subtract in a stated order to get one difference per individual and treat the differences as one sample: x̄d ± t∗ × sd/√n, df = number of differences − 1. 15 students before and after gives df = 14.
What is the sample data condition for a t-interval?
Population approximately normal, or n ≥ 30, or, for n below 30, sample data free from strong skewness and outliers. For pairs, check the differences, not each column.
What conditions apply to a one-sample t-interval?
Randomization: random sample or randomized experiment. 10%: n ≤ 10% of N when sampling without replacement. Sample data: normal population, n ≥ 30, or no strong skew or outliers.

4.3 Justifying a Claim Based on a Confidence Interval for a Population Mean or Population Mean Difference

How do you interpret a t-interval for a population mean?
"We are C% confident that the interval from a to b [units] captures the true mean [variable] of all [population]." It is about the mean, not where 95% of individual values lie.
How do you interpret a matched-pairs interval for μd?
"We are C% confident that the interval from a to b captures the true mean difference in [variable] (first − second) for [population]." State the order of subtraction.
How do you use a t-interval to judge a claimed mean?
Outside: convincing evidence against it. Inside: plausible, not proved. For (12.4, 15.0) cm, 15.5 cm is rejected and 13 cm is plausible.
What does an interval for μd entirely below 0 show?
Convincing evidence of a negative mean difference. With d = after − before, (−2.5, −0.8) points shows the mean score fell. An interval containing 0 gives no convincing evidence of a difference.
How do confidence level and sample size affect a t-interval?
Higher level: t∗ up, so the width grows; SE unchanged. Larger n: SE down and t∗ slightly down, so the width shrinks. n × 9 divides the width by about 3, slightly more because t∗ also falls.
How do you interpret the confidence level of a t-interval?
If we took many random samples of size n and built a C% t-interval from each, about C% of the intervals would capture the true mean.
What scope of conclusion does a t-interval allow?
Generalising to a population needs random sampling; a cause-and-effect conclusion needs random assignment. The margin of error covers random variation only, not bias.

4.4 Setting Up a Test for a Population Mean or Population Mean Difference

How do you write hypotheses for a one-sample t-test?
H₀: μ = μ₀; Hₐ: μ below, above or not equal to μ₀, from the question's wording. Define μ as the true mean [variable, units] of all [population]. "At most 9.0 litres" with a suspicion of more: H₀: μ = 9.0, Hₐ: μ above 9.0.
How do you write hypotheses for a matched-pairs t-test?
H₀: μd = 0; Hₐ: μd below, above or not equal to 0, where μd is the true mean of (A − B) for the population. The direction of Hₐ must match the order of subtraction.
How do you recognise matched-pairs data?
Two values on the same individual (before and after, or both treatments) or natural pairs. Analyse one sample of differences, not two samples. 25 workers timed with old and new tools give 25 differences.
What conditions apply to a one-sample t-test?
Random sample or random assignment (random order of treatments for pairs); 10% when sampling; population approximately normal, or n ≥ 30, or for n below 30 no strong skew and no outliers (in the differences, for pairs).
Is the sample data condition met for n = 18 with an outlier?
No. With n below 30 and no evidence that the population is normal, the sample data must be free from outliers. Do not write "np₀ ≥ 10", which is a proportion check.
Can you test a mean when n is below 30?
Yes, if the population is approximately normal or the sample shows no strong skewness or outliers. Check the data before deciding.

4.5 Carrying Out a Test for a Population Mean or Population Mean Difference

What is the test statistic for a one-sample t-test?
t = (x̄ − μ₀)/(s/√n), with df = n − 1; for pairs, t = (x̄d − 0)/(sd/√n) with n = number of pairs. n = 16, x̄ = 81.5, s = 6, H₀: μ = 78: t = 3.5/1.5 ≈ 2.33, df = 15.
How do you find the p-value for a t-test?
Hₐ with "above": P(T ≥ t). "Below": P(T ≤ t). "Not equal": 2 × P(T ≥ |t|), using a t-distribution with n − 1 df. t = 2.05 with n = 25, one-sided: about 0.0257.
What should you do if your df is not in the t-table?
Use the next smaller df. A t-table gives only a range for the p-value; technology gives a value.
How do you conclude a t-test?
Compare the p-value with α using both numbers: p-value 0.08 with α = 0.05 means fail to reject H₀, so there is not convincing evidence for Hₐ. Conclude about μ or μd for the population, never about x̄.
What does the t statistic measure?
How many standard errors x̄ is from μ₀. If H₀ is true and conditions hold, it follows a t-distribution with n − 1 degrees of freedom.
What does the p-value of a t-test mean?
The probability, assuming H₀ is true, of a test statistic at least as extreme as the observed one in the direction of Hₐ. It is not the probability that H₀ is true.

4.6 Sampling Distributions for the Difference Between Two Sample Means

What are the mean and standard deviation of x̄₁ − x̄₂?
Mean μ₁ − μ₂, so x̄₁ − x̄₂ is unbiased. SD = √(σ₁²/n₁ + σ₂²/n₂): the variances add. σ₁ = 10, n₁ = 25, σ₂ = 6, n₂ = 16: √(4 + 2.25) = 2.5.
When is the sampling distribution of x̄₁ − x̄₂ normal?
Normal if both populations are normal (any n). Approximately normal if the populations are not normal but n₁ ≥ 30 and n₂ ≥ 30 (both, not their sum). Strongly skewed populations with n₁ = 20 fail.
What conditions apply to the sampling distribution of x̄₁ − x̄₂?
Two independent random samples or random assignment; n₁ ≤ 10% of N₁ and n₂ ≤ 10% of N₂ when sampling (not for an experiment); and the normal shape condition.
How does sample size affect the SD of x̄₁ − x̄₂?
Multiplying both sample sizes by k divides the standard deviation by √k.
When should the x̄₁ − x̄₂ formula not be used?
For paired data, where the same individuals are measured twice. The two samples must be independent: different individuals, not linked.
How do you find a z-score for an observed difference in sample means?
z = [(x̄₁ − x̄₂) − (μ₁ − μ₂)] / σ(x̄₁ − x̄₂). Keep the order of subtraction: μ₁ = 40 and μ₂ = 46 give a mean of −6. A small probability gives reason to doubt a claim, not proof.

4.7 Constructing a Confidence Interval for the Difference Between Two Population Means

How do you construct a two-sample t-interval for μ₁ − μ₂?
(x̄₁ − x̄₂) ± t∗ √(s₁²/n₁ + s₂²/n₂). n₁ = n₂ = 20, s₁ = 3, s₂ = 4: SE = √(9/20 + 16/20) ≈ 1.118. Do not add the separate standard errors, and do not choose "pooled".
What degrees of freedom does a two-sample t-interval use?
From technology; always between the smaller of n₁ − 1 and n₂ − 1, and n₁ + n₂ − 2. For n₁ = 12 and n₂ = 18, between 11 and 28. The conservative choice is the smaller of n₁ − 1 and n₂ − 1.
What is the sample data condition for a two-sample t-interval?
Both n ≥ 30, or both populations approximately normal. If either n is below 30, both samples must be free from strong skewness and outliers (dot plot or 1.5 × IQR rule). Check both groups.
When should you use a one-sample t-interval on differences instead of a two-sample interval?
When the data are paired, such as ten patients measured before and after a drug. Use the 10 differences.
How do you get the point estimate and margin of error from a two-sample interval?
Point estimate = (a + b)/2, margin of error = (b − a)/2. (2.4, 9.0) gives 5.7 and 3.3.
What conditions apply to a two-sample t-interval?
Two independent random samples or a randomized experiment; n₁ ≤ 10% N₁ and n₂ ≤ 10% N₂ when sampling (not for an experiment); and the sample data condition for both groups.

4.8 Justifying a Claim Based on a Confidence Interval for the Difference Between Two Population Means

How do you interpret a two-sample t-interval for μ₁ − μ₂?
"We are C% confident that the interval from a to b [units] captures the difference (group 1 minus group 2) in the mean [response] of all [population 1] and all [population 2]."
What does a two-sample interval containing 0 show?
No convincing evidence of a difference, as with (−4.1, 2.7). It does not prove the means are equal. Judge by whether 0 is inside, not how much of the interval is positive.
How do you judge a directional claim with an interval for μ₁ − μ₂?
Every value satisfies the claim: convincing evidence for it. The interval straddles the boundary: not convincing. With (3.5, 9.5), "at least 2 higher" is supported; "more than 4 higher" is not.
What happens to an interval for μ_A − μ_B when the order is reversed?
(a, b) becomes (−b, −a), with the same conclusion. (1.2, 6.8) becomes (−6.8, −1.2) for μ_B − μ_A.
How do you interpret the confidence level for a difference in means?
In repeated random sampling with the same sample sizes from these populations, about C% of intervals built this way would capture μ₁ − μ₂.
What scope of conclusion does a two-sample interval allow?
Random samples allow generalising to the populations; random assignment allows cause-and-effect conclusions. For an experiment, refer to the difference between treatments for subjects like these.

4.9 Setting Up a Test for the Difference Between Two Population Means

How do you write hypotheses for a two-sample t-test?
H₀: μ₁ − μ₂ = 0 (always equality). Hₐ: above 0, below 0, or not equal to 0, from the question's wording. "Different rate of drying" gives Hₐ: μ_new − μ_old ≠ 0. Use μ, never x̄.
Does the two-sample t-test pool the variances?
No. It uses s₁ and s₂ separately and does not assume σ₁ = σ₂.
What sample data condition applies to a two-sample t-test?
Both n ≥ 30, or populations approximately normal, or both samples free from strong skewness and outliers. Samples of 18 and 40 of unknown shape: check both for skewness and outliers, because one is under 30.
How is the 10% condition checked for a two-sample t-test?
Separately for each sample: n₁ ≤ 0.10N₁ and n₂ ≤ 0.10N₂. 45 from 600 passes (limit 60); 45 from 380 fails (limit 38). Not needed in a randomized experiment.
How should μ₁ and μ₂ be defined for a two-sample t-test?
In context: the population (or true) mean of the response variable, with units, for each population or treatment. Pick the order of subtraction once and keep it from definitions to Hₐ.
When is a two-sample t-test the wrong procedure?
When the data are paired (same individuals twice, or natural pairs); use a one-sample t-test on the differences instead.

4.10 Carrying Out a Test for the Difference Between Two Population Means

What is the test statistic for a two-sample t-test?
t = ((x̄₁ − x̄₂) − 0)/SE with SE = √(s₁²/n₁ + s₂²/n₂): how many standard errors the difference is from 0. x̄₁ − x̄₂ = 3.0 and SE = 1.2 give t = 2.5.
What degrees of freedom does a two-sample t-test use?
From technology, between min(n₁ − 1, n₂ − 1) and n₁ + n₂ − 2. The conservative df, min(n₁ − 1, n₂ − 1), gives a slightly larger p-value. Samples of 15 and 22: 14 and 35. Do not use n₁ + n₂ − 2 as exact.
How do you find the p-value for a two-sample t-test?
Hₐ: μ₁ − μ₂ above 0: P(t ≥ observed t). Below 0: P(t ≤ observed t). Not equal: 2 × P(t ≥ |observed t|). Compare the p-value, not t, with α.
How do you interpret the p-value of a two-sample t-test?
"Assuming the true means of group 1 and group 2 are equal, there is a [p] probability of getting a difference in sample means of [observed] or more extreme, in the direction of Hₐ, by chance alone."
How do you conclude a two-sample t-test?
p-value 0.074 is greater than α = 0.05, so fail to reject H₀: not convincing evidence that the population means differ. Never "accept H₀" or "the means are equal".
When can a two-sample t-test support a causal claim?
Only with random assignment in an experiment. Random samples allow generalising to the populations but not cause and effect.

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