Inference for Means (C23-C25 BVD)

*AP Statistics Review

*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*df(Sx/sqrt(n))

*Sample statistic +/- ME

*ME = # standard errors reaching out from statistic.

*T-interval on calculator

*Finding the critical value (t

star)

*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)

*Finding Sample Size

*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.

*Conditions/ Assumptions to

Check

*For 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

*Hypothesis Test for 1 mean

*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.

*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.

*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