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10-1 to 10-2 Quiz THURSDAY 2-13-14
HW-pg. 657-658 (10.36)
**READ 10.2 Summary on pgs. 658-659**
2-11-14
Warm up—AP STATS
How do we calculate the standard deviation of
the sampling distribution of x , a) when we do
know, and b) when we don’t know the standard
deviation of the population? HINT-Think about
the CI formula in both cases.
Name____________________________
Date__________
AP Stats
10.2 Estimating a Population Mean Day 2
Objectives:

Describe what is meant by paired t procedures.

Calculate a level C t confidence interval for a set of paired data.

Explain what is meant by a robust inference procedure and comment on the
robustness of t procedures.

Discuss how sample size affects the usefulness of t procedures.
Comparative studies are more convincing then single-sample investigations. For that
reason, one-sample inference (what we have done so far in chapter 10) is less common
than comparative inference. A common design to compare two treatments makes use of
one-sample procedures. Recall from Chapter 5 that in a matched pairs design, subjects
are matched in pairs and each treatment is given to one subject in each pair.
Alternatively, each subject receives both treatments is some order. A coin toss can be
used to assign the treatments to the two subjects in each pair or to determine the order
in which an individual subject receives the two treatments. Another situation calling for
__________ ___ _______________ is before-and-after observations on the same
subject.
Paired t Procedures
To compute the responses to the ________ treatments in a matched pairs design or
before-and-after measurements on the same ______________, apply one-sample t
procedures to the observed ________________.
The parameter µ in a paired t procedure is:
 the mean difference in the responses to the ______ treatments within
matched pairs of subjects in the entire population (when subjects are matched
in pairs) or
 the mean difference in response to the two treatments for individuals in the
population (when the same subject receives ________ treatments) or
 the mean difference between _________-and-__________ measurements for
all individuals in the population (for before-and-after observations for the same
___________________.)
The proper analysis depends on the design used to produce the data.
See Example 10.10 (pp.651-653) This example is not a before-and-after type problem
but is a matched pair problem because the same individual received both treatment. It
shows how to turn paired data into single-sample date by taking differences within each
pair.
Many studies that require the use of paired t procedures involve individuals who are not
chosen at __________ from the population of interest because the purpose is often to
compare two treatments rather than to _______________ to a larger population. In
such cases we may not be able to generalize our findings to the population of
____________.
Make sure you understand the different purposes of random selection and random
assignment.
Random selection of individuals for a statistical study allows us to _______________
the results of that study to a larger population. Random assignment of treatments to
subjects in an experiment lets us investigate whether there is evidence of a treatment
__________, which might suggest that the treatment caused the observed difference
(allows us to ____________ treatments).
The t confidence interval is exactly correct when the distribution of the population is
exactly ___________. No real data are exactly Normal. The usefulness of the t
procedures in practice therefore depends on how strongly they are affected by lack of
Normality. Procedures that are not strongly affected are called robust. The t
procedures are not robust against ___________, because x and s are not resistant to
outliers.
Robust Procedures
An inference procedure is called robust if the probability calculations involved in that
procedure remain fairly ___________________ when a condition for use of the
procedure is _______________. For confidence intervals, this means that the stated
confidence level is still fairly accurate.
Fortunately, the t procedures are quite _______________ against non________________ of the population when there are no _________________,
especially when the distribution is roughly symmetric. Larger samples improve the
accuracy of critical values from the t distributions when the population is not
_____________. The main reason for this is the ____________ _____________
_______________. The t statistic is based on the sample mean x , which becomes more
nearly ______________ as the sample size gets ________________ even when the
population does not have a _________________ distribution.
Always make a plot (Npp best) to check for skewness and outliers before you use
the t procedures for small samples.
Here are practical guidelines for inference on a single mean.
Using the t Procedures
 Except in the case of small samples, the assumption that the data are an _____
from the population of interest is more important than the assumption that the
population distribution is ________________.
 Sample size less than 15. Use ___ procedures if the data are close to Normal. If
the data are clearly non-Normal or if ____________ are present, do not use t
procedures.
 Sample size at least 15. The ___ procedures can be used except in the presence
of ________________ or strong _________________.
 Large samples. The ___ procedures can be used even for clearly ____________
distributions when the sample is large, say n  30
*If your sample data would give a biased estimate for some reason, then you should not
bother computing a t interval. Or if the data you have is the entire population of
interest, then there is no need to perform inference.
*Use of t procedures is warranted any time inference is done on a population mean
without knowledge of σ (which is almost always)!
10.33 Does playing the piano make you smarter?
Do piano lessons improve the spatial-temporal reasoning of preschool children?
Neurobiological arguments suggest that this may be true. A study designed to test this
hypothesis measured the spatial-temporal reasoning of 34 preschool children before and
after six months of piano lessons. The changes in the reasoning scores are:
2
5
7
-2
2
7
4
1
0
7
3
4
3
4
9
4
5
2
9
6
0
3
6
-1
3
4
6
7
-2
7
-3
4
4
a)
Construct and interpret a 95% CI for the mean improvement in reasoning scores.
b)
Can we conclude that playing the piano causes an improvement in spatial-temporal
reasoning in children? Justify your answer.
Parameter
Conditions
SRS
Normality
Independence
Calculations
Interpretation