7.1-7.9

Interpreting P-value (Significance tests)

Assuming (context of Ho), there is a (percent) probability of getting a sample (proportion/mean) of (p/mu) or (more/less) purely by chance.

Conclusion (Significance tests)

Because (p-value) is (less than/greater than) (alpha), we (reject/fail to reject) the Ho. We (have/do not have) convincing evidence of (Ho in context).

p-value is less than significance level…

significant → reject null

p-value is greater than significance level…

not significant → fail to reject the null

Interpreting Power

If the true (mean/difference in mean) of (context) is (sample mean), there is a (Power) probability of correctly rejecting the null of (Ho).

Type I Error

Truth: H0 true, Conclusion: Reject H0 P(Type I)=Alpha (Significance Level)

Type II Error

Truth: Ha true, H0 false, Conclusion: Fail to reject H0

Consequences…

Health consequences always considered worse

Power

Truth: Ha true, Conclusion: Reject H0 P(Reject H0 I Ha is true) P(Power)=1-P(Type II)

Increase Power by:

increasing n, increasing alpha, increasing distance btwn H0 & Ha

Interpreting CI

We are (conf level)% confident the interval from (A) to (B) captures the true (mean/proportion) of (context).

Interpreting CL

If we make many (conf level)% confidence intervals, we expect about (conf level)% to capture the true (mean/proportion) of (context).

A CI gives…

plausible values

normal/z-score smaller than t…

larger MOE for interval

for critical values the MOE of error is smaller and confidence interval is smaller

Conf. Int. for Mu 1) state:

parameter (true mean…), confidence **level**

Conf. Int. for Mu 2) plan:

1 sample t-interval, 3 conditions

a) random sample/assignment

b) 10% condition

c) normality (meets one of three options)

• population is normal → sample distribution is normal

• CLT n >= 30

• graph sample data for skew/outliers

Conf. Int. for Mu 3) do:

x-bar +- t\* sx/root n (substitute #s in), interval

t\* → table B w/ tail % & df=n-1

If df not in table, round down

Conf. Int. for Mu 4) conclude:

we are _% confident…

choosing sample size → t *unknown if n unknown, so use z*\* instead

Conf. Int. for Difference in Means 1) state:

true difference in means, confidence **level**

Conf. Int. for Difference in Means 2) plan:

2 sample t-int for Mu1-Mu2

3 conditions (x2)

Conf. Int. for Difference in Means 3) do:

(xbar1-xbar2)+-t\*root(s1^2/n1+s2^2/n2)

Conf. Int. for Difference in Means 4) conclude:

if 2 diff. n’s, use smaller n for df/t\* → gives a conservative estimate

real df is bigger 2-sample (show formula)

(-,-)

x-bar2 is greater

(+,+)

x-bar1 is greater

(-,+)

no difference

one sample t-int for Mdiff

x-bar diff+- t\* sdiff/rootn

mean diff → 1 mean = 1 sample

paired data

2 specific data points that must be paired together (usually because they’re both from 1 individual)

Test statistic (t-score)

t = x-bar - Mu/(Sx/rootn)

Significance Test for Mu and CI

calculate test statistic (t-score) and use table B w/ df & tail probability OR tcdf(lower, upper, df)

**OR**

calculator → copy title up to x-bar

If H0 in int.

H0 plausible → fail to reject

If H0 not in int.

H0 not plausible → reject

For 2-sided sig tests ONLY

a c% confidence int. will make the same decision as alpha = 1-C%

Significance Test for a Difference in Means

t-score = (xbar1-xbar2)-(Mu1-Mu2 from H0) / root(s1^1/n1+s2^2/n2)

& graph & tcdf(labeled)

**OR**

2 sample t-test → copy title up to x

paired data →

1 sample t-test for Mudiff

t=xbardiff - Mudiff / sdiff/rootn

1 sample

• matched pairs creating 1 sample

• subtract then average

• 1 mean (Mudiff)

2 samples

• 2 groups and 2 means

• average then subtract

• difference of means (Mu1-Mu2)