Chapter 11Inference for Distributions

AP Statistics

11.1 – Inference for the Mean of a Population

σ is unknown

• Often the case in practice• When σ is known

• When σ is unknown• Whoa, way different!

n

xz

n

sx

t

Really just

x

Becomes

This is just Standard error of the Sampling mean

Standard Error

n

n

s

x

• Cousins of the z-distribution (Normal)

• Conditions for inference about a mean– Random? – to generalize about the population– Normal? – Verify if the sampling distribution about the

mean is approximately normal.– N>=10n? - Independent?

• t(k) distribution where k = n – 1 degrees of freedom– S has n-1 degrees of freedom

t-distributions

x

• Similar to Normal curve; symmetric, single peaked, bell shaped

• Spread of t-dist. is greater than z-dist.• As degrees of freedom increase, the t(k) density

curve approaches the normal curve more closely.– (s estimates more accurately as n increases)

• t* uses upper tail probabilities (look at table)• Y1=normalpdf(x)• Y2=tpdf(x,df)

t(k) distributions

Using the t* table

• What critical value t* would you use for a t distribution with 18 degrees of freedom having probability 0.9 to the left of t*?

Using the t* table

• What t* value would you use to construct a 95% confidence interval with mean and an SRS of n = 12?

Using the t* table

• What t* value would you use to construct a 80% confidence interval with mean and an SRS of n = 56?

t-CI’s & t-tests

• 1-sample t-interval VS. 1-sample t-test

Normal Probability Plot

T(9) distribution

Data Software Packages

Matched Pairs t-procedures

• Comparative Studies are more convincing than single-sample investigations

• To compare the responses of the two treatments in a matched pairs design, apply the 1-sample t-procedures to the Observed DIFFERENCES!

Statistical Software Packages (con’t)

Robustness of t-procedures

• A CI or Significance Test is called robust if the confidence level or P-value does not change very much when assumptions of the procedure or violated.

• Outliers? – Like and s, the t-procedures are strongly influenced by outliers.

x

Quite Robust when No Outliers

Sample size increases CLT more robust!

Using the t-procedure

• SRS is more important than normal (except in the case of small samples)

• n < 15, use t-procedures if the data are close to normal

• n ≥ 15, use t-procedures except in presence of outliers or strong skewness

• Large samples (roughly n ≥ 40), t-procedures can be used even for clearly skewed distributions

• p. 636-637 - histograms

The power of the t-test

• Power measures ability to detect deviations from the null hypothesis Ho

• Higher power of a test is important!

• Usually assume a fixed level of significance, α = 0.05

Here we go again. . . Power!

• Director hopes that n=20 teachers will detect an average improvement of 2 pts in the mean listening score. Is this realistic?

• Hypotheses? • Test against the alternative =2 when n=20.• Impt: Must have a rough guess of the size of to

compute power! = 3 (from past samples)