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Chapter 23: Inferences About Means
Central Limit Theorem: (pg. 521, Ch. 18) No matter what population the random
sample comes from, the shape of the sampling distribution is approximately
Normal as long as the sample size is large enough. The larger the sample used,
the more closely the Normal approximates the sampling distribution.
When creating a sampling distribution, we need:
1. a random sample of quantitative data
2. the true population standard deviation, 
If we don’t have  (which is almost always!), we have to estimate it, using the
standard error:
SE ( y ) 
HOWEVER, there is additional variance between samples that must be
accounted for…
Gosset’s t: (pg. 522) accounts for additional variance based on sample size
use the t model when you only know s, the sample standard
deviation (not  , the population standard deviation)…this is
almost always true!
t distributions are always bell shaped, but change with sample
t model is often denoted: t df
degrees of freedom: df = n-1
TI Tips – pg. 524-525
Confidence Intervals:
Assumptions and Conditions
o Independence Assumption
 Randomization Condition
 10% Condition
o Normal Population Assumption – NEW!
 Nearly Normal Condition: the data come from a distribution
that is unimodal and symmetric
*make a histogram or Normal probability plot to check*
 for small sample sizes (n<15): the data should follow
the Normal model pretty closely
 for moderate sample sizes (15<n<40): the t methods
will work well as long as the data are unimodal and
reasonably symmetric
large sample size (n>40): t methods are safe to use
even if the data are skewed
*make a histogram anyway to check for outliers
*if the data has multiple modes, it may need to be
separated into different categories
One-Sample t-interval
When the conditions are met, we are ready to find the confidence interval for the
population mean,  . The confidence interval is:
y  t *n1 SE ( y )
Where the standard error of the mean, SE ( y ) 
The critical value t *n1 depends on the particular confidence level, C, that you specify
and on the number of degrees of freedom, n-1, which we get from the sample size.
**Just Checking: pg. 526-527
**Step-by-Step: pg. 527-529
**TI Tips: pg. 529-530
One-sample t-test for the mean
The conditions for the one-sample t-test for the mean are the same as for the onesample t-interval. We test the hypothesis Ho :  o using the statistic
tn  1 
The standard error of y is SE ( y ) 
y  o
SE ( y )
When the conditions are met and the null hypothesis is true, this statistic follows a
Student’s t-model with n-1 degrees of freedom. We use that model to obtain a P-value.
Steps for a One-sample t-test:
Hypotheses: This time use  , not y . (Remember for proportions we
used p , not p̂ .
Model: Check the conditions.
Mechanics: With conditions complete, calculate t, draw the curve
(indicating degrees of freedom), shade the region representing the Pvalue, and find P (using technology or a t table for critical values).
Conclusion: Link p-value to the decision in context. Be sure the
conclusion talks about the mean of a population.
*If you have an outlier, perform the analysis twice.
**Step-by-Step: pg. 531-533
**TI Tips: pg. 533
“Significance and Importance”: When performing a hypothesis test, use the CI to
determine possible values and see if the conclusion is important (“statistically significant”
does not necessarily mean important)
**Just Checking: pg. 534
Intervals and Tests:
A level C confidence interval contains all of the plausible null hypothesis
values that would not be rejected by a two-sided hypothesis test at
alpha level 1 – C
Sample Size calculations:
ME  t * n  1
s is usually unknown
n is unknown (it’s what we’re looking for)
s can be estimated (if we have no idea, we can take a small “pilot
use z* instead of t* to get an estimate
your sample size calculations (margin of error) won’t be exact
#2, 4, 5, 6, 10, 11, 12, 17
#19, 21, 26, 29, 33, 34