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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.

- 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.

X-bar +/- t*df(Sx/sqrt(n))

- Sample statistic +/- ME
- ME = # standard errors reaching out from statistic.
- T-interval on calculator

Draw or imagine a normal model with C% shaded, symmetric about the center.

- What percent is left in the two tails?
- What percentile is the upper or lower fence at?
- Look up that percentile in t-table to read off t(or use invt(.95,df) or whatever percent is appropriate)

ME = t*(SE)

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 SizeFor inference for means check:

- 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

Null: µ is hypothesized value

- 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.

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.

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

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