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AP Statistics – 10.2 Estimating a Population Mean - t Distribution
Confidence Intervals with an unknown population standard deviation
I.
Conditions for Inference about a Population Mean with Unknown Standard Deviation
a. SRS
b. Normality
c. Independence – N has to be at least 10 times the size of your sample for dependent trials
II.
Standard Error
a. When we do not know the standard deviation of the population (often in real practice) we use
the sample standard deviation to estimate the population standard deviation. We will then use
the sample standard deviation in calculating the standard deviation in the sampling distribution.
b. When the standard deviation of a statistic is estimated from the data, the results is called the
s
standard error of the statistic. The standard error of the sample mean is
.
n
III.
t – distribution
a. Use the t distribution when the population standard deviation is unknown.
b. Each sample size has a different t distribution.
c. Each t distribution is identified with degrees of freedom: df = n – 1.
d. The standard Normal distribution is the z distribution.
e. Symmetric about zero, single peaked, and bell-shaped.
f. Larger spread than z distribution (estimating the standard deviation introduces more variability)
g. More area under the tails.
h. As the degrees of freedom increases, the t distribution approaches the z distribution (standard
normal distribution). This happens because as n increases, the estimated standard deviation
becomes closer to the actual standard deviation.
i. t c = critical value for confidence level c
IV.
One-Sample t Confidence Intervals
a. Draw a SRS of size n from a population having unknown mean µ . A level C confidence
s
interval for µ is x ± t
where t is the critical value for the t distribution.
n
b. The interval is exactly correct when the population distribution is Normal and approximately
correct for large n in other cases (CLT).
c. Steps for Confidence Intervals
i. Parameter, Conditions (Random, Normality, Independence), Calculations,
Interpretations.
ii. When the actual df does not appear in Table C, use the greatest df available that is less
than your desired df.
V.
Paired t Procedures
a. To compare the responses to the two treatments in a matched pairs design or before-and-after
measurements on the same subjects, apply one-sample t procedures to the observed differences.
b. The parameter mu in a paired t procedure is…
i. The mean difference in the responses to the two treatments within matched pairs design
of subjects in the entire population (when subjects are matched in pairs)
ii. The mean difference in response to the two treatments for individuals in the population
(when the same subject receives both treatments)
iii. The mean difference between before-and-after measurements for all individuals in the
population (for before-and-after observation on the same individuals).
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VI.
Robustness of t Procedures
a. The usefulness of the t procedures in practice depends on how strongly they are affected by lack
of Normality. Procedures that are not strongly affected are call robust.
b. An inference procedure is called robust if the probability calculations involved remain fairly
accurate when a condition is violated. For confidence intervals, this means that the stated
confidence level is still fairly accurate.
VII.
Using the t Procedures
a. Except in small samples, the assumption that the data are an SRS from the population is more
important than the assumption that the population is Normal.
b. Sample size less than 15: Use t procedure if the data are close to Normal. If there data are
clearly non-Normal of it outliers are present, do not use t procedures.
c. Sample size at least 15: The t procedures can be used except in the presence of outliers or strong
skewness.
d. Large samples: The t procedures can be used even for clearly skewed distributions when the
sample is large (30 or more).
e. Random selection of individuals for a statistical study allows us to generalize the results of that
study to a larger population.
f. Random assignment of treatments to subjects in an experiment lest us investigate whether
there is evidence of a treatment effect, which might suggest that the treatment caused the
observed difference.
VIII.
Cautions
a. When the actual df does not appear in the table, use the greatest df available that is less than
your desired df.
b. Be sure you understand the different purposes of random selection and random assignment.
c. The proper analysis depends on the design used to produce the data.