Stats 2B03: Statistical Methods for Science
Chapter 6: Estimation
6.2 Confidence Interval For A Population Mean
- Sampling distributions and estimations:
- Interval estimate components:
Reliability coefficient: z
⁄
- Interpreting confidence intervals:
Probabilistic interpretation: in repeated sampling, from a normally
distributed population with a known standard deviation, 100(1 – )
percent of all intervals of the form ⁄ will in the
long run include the population mean
Confidence coefficient:
Confidence interval:
⁄
Practical interpretation: when sampling is from a normally distributed
population with known standard deviation, we are 100(
percent confident that the single computed interval
, contains the population mean
⁄
- Precision: quantity obtained by multiplying the reliability factor by the
standard error of the mean. Also called the margin of error
- Sampling from nonnormal populations: for large samples, the sampling
population distribution of is approximately normally distributed.
- Computer analysis
- Alternative estimates of central tendency: median is sometimes preferred
over the mean as a measure of central tendency when outliers are present
- Trimmed mean: estimators that are insensitive to outliers are called robust
estimators. Another robust measure and estimator of central tendency is the
trimmed mean:
Order the measurements
Discard the smallest 100 percent and the largest 100 percent of the
measurements. The recommended value of is something between .1
and .2
Compute the arithmetic mean of the remaining measurements
6.3 The t Distribution
- σ is unknown so we use the sample standard deviation
- √
- Properties of the t distribution:
Has a mean of 0
Symmetrical about the mean
In general it has a variance greater than 1, but the variance
approaches 1 as the sample size becomes large. For df>2, the variance
of the t distribution is df/(df – 2), where df is the degrees of freedom. Alternatively, since here df = n – 1 for n>3, we may write the variance
of the t distribution as (n – 1)/(n – 3)
The variable t ranges from -∞ to +∞.
The t distribution is really a family of distributions, since there is a
different distribution for each sample value of n – 1, the divisor used
in computing s . We recall that n – 1 is referred to as degrees of
freedom.
Compared to the normal distribution, the t distribution is less peaked
in the center and has thicker tails.
The t distribution approaches the normal distribution as n – 1
approaches infinity
- Confidence intervals using T:
When sampling is from a normal distribution whose standard
deviation, σ, is unknown, the 100(1 – α) percent confidence interval
for the population mean, μ, is given by:

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