Inference for Quantitative Data: Means

Ten topics that run Unit 3's machinery again, on quantitative data, and the differences are where the points are. The sampling distribution of x̄ and its spread, why estimating σ with s forces the t distribution and its heavier tails, confidence intervals and tests for a population mean, paired data and why it is a one-sample problem in disguise, and two-sample inference for a difference of means, where variances add even though the means subtract.

Exam weight 10-20%10 topics

Topics

4.1

Sampling Distributions for Sample Means

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

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

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

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

Sampling Distributions for the Difference Between Two Sample Means

Constructing a Confidence Interval for the Difference Between Two Population Means

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

Setting Up a Test for the Difference Between Two Population Means

Carrying Out a Test for the Difference Between Two Population Means

Lesson

Diagnostic

after diagnostic

Key forms For every problem in this unit

Center of x̄

μ: the sample mean is an unbiased estimator of the population mean

Spread of x̄

σ / √n. Divide by ROOT n, and this is the spread of the STATISTIC, not of the data

Why t, not z

σ is unknown, so s stands in. That extra uncertainty is exactly what t charges for

t distribution

heavier tails than z, so critical values sit FARTHER out. It approaches z as df grows

Degrees of freedom

n − 1 for one sample or for paired differences

Conditions

randomization; 10% when sampling without replacement; normal/large sample (plot the data, or n large enough for the CLT)

One-sample t interval

x̄ ± t* · s / √n

Margin of error

t* · s / √n. It covers SAMPLING variability only, never a bad question or nonresponse

One-sample t statistic

t = (x̄ − μ₀) / (s / √n)

Paired data

compute the DIFFERENCES first, then run one-sample t on them. df = (number of pairs) − 1

Interpreting the interval

plausible values for μ, in context. Not for individuals, not for future sample means

Paired or two-sample?

decide from the DESIGN. Same subjects measured twice, or matched pairs, is paired; two separate groups is two-sample

Center of x̄₁ − x̄₂

μ₁ − μ₂

Spread of a difference

√(σ₁²/n₁ + σ₂²/n₂): VARIANCES ADD, even for a difference. Never subtract them, never add SDs

Two-sample t interval

(x̄₁ − x̄₂) ± t* · √(s₁²/n₁ + s₂²/n₂)

Two-sample t statistic

t = (x̄₁ − x̄₂ − 0) / √(s₁²/n₁ + s₂²/n₂)

Hypotheses

about μ₁ − μ₂ (usually H₀: μ₁ − μ₂ = 0), never about x̄

Zero in the interval

no difference is PLAUSIBLE. That is not proof the means are equal

Subtraction order

fix it once and keep it. A negative interval means the SECOND group is larger

Conclusion

compare the p-value to α explicitly, then state the decision IN CONTEXT, with "evidence", never "proof"

Robustness

t procedures tolerate mild skew at decent n, but an outlier in a small sample is a real problem. Look at the data

Unit 4 tools

Challenge bank

1 / 60

60 open-ended problems.

Read the question, work it out, then flip the card to compare your reasoning to the worked solution. Mark each card so you can return to the ones that still bite.

Question

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Worked solution

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Cumulative assessment

Test the unit.

Twenty mixed items drawn from across all 10 topics, with guaranteed misconception-code coverage. Identifies which misconceptions still bite when you cannot see which topic the question came from.

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Check what stuck.

Units 1 through 4, drawn evenly so earlier units get the same share as this one. Twenty questions or a full 42-question section, your choice. Even coverage means this is a retention check rather than a score estimate.

20 or 42questions

50topics

56codes covered

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