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CHAPTER 17 –THINKING ABOUT INFERENCE
17.1 - Conditions for inference about a mean
 We need a simple random sample
 The population must be MUCH larger than the sample, say at least 20 times as large
 Shape of population
o A normal population is not needed for large samples – ( CLT says that x-bar is approximately normal
for large n)
o For small samples, it is enough that the distribution be symmetric and single peaked unless the
sample is very small.
 Both, mu and sigma are unknown parameters
17.2 - The t-distribution
 Symmetric about the mean
 The distribution depends on the degrees of freedom – DF = n - 1
 Bell shaped, but thicker tails and lower in the middle when you compare it with the z-distribution (SND)
 As the degrees of freedom increase, the t distribution approaches the standard normal distribution (SND)
 Mean is zero
 Standard deviation is more than 1

x
s
n

T statistic is t 

Finding t-scores using the t-table
o For constructing confidence intervals
o For testing hypothesis
17.3 - The one-sample t-confidence interval
 The interval is exact when the population is normal and is approximately correct for large n in other cases
 Construct with the calculator 8:T Interval

Construct by hand using the formula:

Interpret the results
xt*
s
n
17.4 - The one-sample t-test
 Write hypothesis
 Sketch graph, label and shade
 Run the test in the calculator 2:TTest
 Use the results to write the conclusion
By hand:

Find the test statistic t 

Write the conclusion
x
and use the t-table to find an interval for the p-value
s
n
17.5 Using technology
17.6 – Matched pairs – t-procedures
 Used to compare responses to the two treatments in a matched pairs design
o In some cases, each subject receives both treatments in a random order
o In others, the subjects are matched in pairs as closely as possible, and each subject in a pair receives
one of the treatments.
 Find the difference between the responses within each pair
 Apply the one-sample t procedures to these differences
o Read example 17.4 page 455
17.7 – Robustness of the t-procedure
A confidence interval or significance test is called robust if the confidence level or P-value does not change very
much when the conditions for use of the procedure are violated.

Except in the case of small samples, the condition that the data are an SRS from the population of interest
is more important than the condition that the population distribution is Normal.

Sample size less than 15: Use t procedures if the data appear close to Normal (roughly symmetric, single
peak, no outliers). If the data are clearly skewed or if outliers are present, do not use t.

Sample size at least 15: The t procedures can be used except in the presence of outliers or strong
skewness.

Large samples: The t procedures can be used even for clearly skewed distributions when the sample is
large, roughly n ≥ 40.
SUMMARY


Tests and confidence intervals for the mean µ of a Normal population are based on the sample mean
of an SRS. Because of the central limit theorem, the resulting procedures are approximately correct for other
population distributions when the sample is large.
The standardized sample mean is the one-sample z statistic
If we knew σ, we would use the z statistic and the standard Normal distribution.

In practice, we do not know σ. Replace the standard deviation
of
by the standard error
to
get the one-sample t statistic
The t statistic has the t distribution with n − 1 degrees of freedom.


There is a t distribution for every positive degrees of freedom. All are symmetric distributions similar in
shape to the standard Normal distribution. The t distribution approaches the N(0, 1) distribution as the
degrees of freedom increase.
A level C confidence interval for the mean µ of a Normal population is
The critical value t* is chosen so that the t curve with n – 1 degrees of freedom has area C between –t* and t*.

Significance tests for H0: µ = µ0 are based on the t statistic. Use P-values or fixed significance levels
from the t(n − 1) distribution.

Use these one-sample procedures to analyze matched pairs data by first taking the difference within each
matched pair to produce a single sample.

The t procedures are quite robust when the population is non-Normal, especially for larger sample sizes.
The t procedures are useful for non-Normal data when n ≥ 15 unless the data show outliers or strong skewness.