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AP Statistics Review. Inference for Means (C23-C25 BVD ). Unless the standard deviation of a population is known, a normal model is not appropriate for inference about means. Instead, the appropriate model is called a t-distribution .

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Ap statistics review

AP Statistics Review

Inference for Means (C23-C25 BVD)

T distributions

  • Unless the standard deviation of a population is known, a normal model is not appropriate for inference about means. Instead, the appropriate model is called a t-distribution.

  • T-distributions are unimodal and symmetric like Normal models, but they are fatter in the tails. The smaller the sample size, the fatter the tail.

  • In the limit as n goes to infinity, the t-distribution goes to normal.

  • Degrees of freedom (n-1) are used to specify which t-distribution is used.

  • T-table only has t-scores for certain df, and the most common C/alphas. If using table and desired value is not shown, tell what it would be between, or err on the side of caution (choose more conservative df, etc.)

  • Use technology to avoid the pitfalls of the table when possible.


Confidence interval for 1 mean

  • X-bar +/- t* normal model is not appropriate for inference about means. Instead, the appropriate model is called a df(Sx/sqrt(n))

  • Sample statistic +/- ME

  • ME = # standard errors reaching out from statistic.

  • T-interval on calculator

Confidence Interval for 1 Mean

Finding the critical value t star

Finding the critical value (t star)

Finding sample size

  • ME = t*(SE) about the center.

    Plug in desired ME (like within 5 inches means ME = 5).

    Plug in z* for desired level of confidence (you can’t use t* because you don’t know df).

    Plug in standard deviation (from a sample or a believed true value, etc. Solve equation for n.

Finding Sample Size

Conditions assumptions to check

  • For inference for means check: about the center.

  • 1. Random sampling/assignment?

  • 2. Sample less than 10% of population?

  • 3. Nearly Normal? – sample size is >30 or sketch histogram and say could have come from a Normal population.

  • 4. Independent – check if comparing means or working with paired means

  • 5. Paired - check if data are paired if you have two lists

Conditions/ Assumptions to Check

Hypothesis test for 1 mean

  • Null: µ is hypothesized value about the center.

  • Alternate: isn’t, is greater, is less than

  • Hypothesized Model: centers at µ, has a standard deviation of s/sqrt(n)

  • Find t-score of sample value using n-1 for df

  • Use table or tcdf to find area of shaded region. (double for two-tail test).

  • T-test on calculator– report t, df and p-value.

Hypothesis Test for 1 mean

Inference for 2 means

  • If data are paired, subtract higher list – lower list to create a new list, then do t-test/t-interval.

  • If data are not paired:

  • Check Nearly Normal for both groups – both must individually be over 30 or you have to sketch each group’s histogram and say could’ve come from normal population

  • CI: mean1-mean2 +/- ME --- use calculator because finding df (and therefore also t*) is rather complicated.

  • SE for unpaired means is sqrt(s12/n1 + s22/n2)

  • If calculator asks “pooled” – choose “No”.

  • Null for paired: µd = 0 (usually)

  • Null for unpaired: µ1 - µ2 = 0

  • Don’t forget to define variables.

  • Use 2-Sample T-Test and 2-Sample T-Interval in calculator for data that are not paired.

Inference for 2 Means

What to write

  • State: name of test, hypothesis if a test, alpha level if a test, define variables

  • Plan: check all conditions – check marks and “yes” not good enough

  • Do: interval for intervals, test statistic, df (if appropriate) and p-value for tests It is good to write the sample difference if doing inference for two proportions or two means, but make sure no undefined variables are used

  • Conclude: Interpret Confidence Interval or Hypothesis Test – See last slide show for what to say

What to Write