Transcript Descriptive Intervals

Confidence Intervals
General Mean ()
Computation
First, edit and summarise the data. Obtain: sample
size (n), sample mean (m) and sample standard
deviation(sd).
Note the indicated confidence level for the desired
interval. Obtain the confidence coefficient (Z) via table
look-up.
Compute the interval as lower = m ─ Z*(sd/n)
And upper = m + Z*(sd/n). Write the interval as
[lower,upper].
Success Rate for
corresponding
Family of Intervals
“Central
Probability
Mass”
ProbCent
“Left
Tail
Probability
Mass”
ProbLt
-Z
0
“Right
Tail
Probability
Mass”
ProbRt
Z
Confidence Multiplier
Confidence Coefficient Examples
Using the table linked here, find coefficients for
confidence levels 50%, 70%, 80%, 90%, 99%.
Z ProbRT ProbCent
50%: 0.70 0.24196 0.51607; Z ≈ .70
70%: 1.05 0.14686 0.70628; Z ≈ 1.05
80%: 1.30 0.09680 0.80640; Z ≈ 1.30
90%: 1.65 0.049471 0.90106; Z ≈ 1.65
99%: 2.60 0.004661 0.99068 ; Z ≈ 2.60
Interpretive Base: General Mean
Briefly identify the population of interest.
Briefly identify the population mean of
interest.
Briefly describe the family of samples.
Briefly describe the family of intervals.
Apply confidence level to the family of
intervals.
Interpret the computed interval.
Populations, Samples and Families 1
We begin with a population T and a population mean . For any
fixed sample size, n, the Family of Samples consists of the
collection of all possible random samples of size n from T. Each
individual member of this Family of Samples is a single random
sample of size n from T.
Family of Sample Means: A sample mean, m, can be computed
from each member of the Family of Samples. If we compute a
sample mean from each member of the Family of Samples, we
obtain a Family of Means. Each member of this Family is a
single sample mean computed from a member of the Family of
Samples.
When the Family of Sample Means is based on large (n>30)
random samples from a population T, the members of the
Family of Sample Means tend to cluster around the population
mean  for T. That is, a large proportion of the members of the
Family of Sample Means are relatively close to the population
mean  for T.
Populations, Samples and Families 2
Family of Intervals (for the Mean): For a fixed multiplier
Z, an interval, m ± Z(sd/n), can be computed from
each member of the Family of Samples. If we compute
an interval from each member of the Family of
Samples, we obtain a Family of Intervals. Each
member of this Family is a single interval of the form
m ± Z(sd/n) computed from a member of the Family
of Samples.
When the Family of Intervals is based on large (n>30)
samples, a fixed percentage of the members of the
Family of Intervals contain the population mean  for
T. The actual value of this percentage depends on the
multiplier Z.
Proportion (P)
Computation
First, edit and summarise the data. Obtain: sample size (n)
and sample event count (nE).
Compute p = (nE/n) and sdp = ( p*(1 ─ p) / n).
Note the indicated confidence level for the desired interval.
Obtain the confidence coefficient (Z) via table look-up.
Compute the interval as lower = p ─ (Z*sdp)
And upper = p + (Z*sdp). Write the interval as [lower,upper].
Interpretive Base: Proportion
Briefly identify the population of interest.
Briefly identify the population proportion of
interest.
Briefly describe the family of samples.
Briefly describe the family of intervals.
Apply confidence level to the family of
intervals.
Interpret the computed interval.
Populations, Samples and Families 1
We begin with a population T and a population proportion P. For
any fixed sample size, n, the Family of Samples consists of the
collection of all possible random samples of size n from T. Each
individual member of this Family of Samples is a single random
sample of size n from T.
Family of Sample Proportions: A sample proportion, p, can be
computed from each member of the Family of Samples. If we
compute a sample proportion from each member of the Family
of Samples, we obtain a Family of Proportions. Each member
of this Family is a single sample proportion computed from a
member of the Family of Samples.
When the Family of Sample Proportions is based on large (n>30)
random samples from T, the members of the Family of Sample
Proportions tend to cluster around the population proportion P.
That is, a large proportion of the members of the Family of
Sample Proportions are relatively close to the population
proportion P.
Populations, Samples and Families 2
Family of Intervals (for Proportion): For a fixed multiplier
Z, an interval, p ± (Z*sdp), can be computed from
each member of the Family of Samples. If we compute
an interval from each member of the Family of
Samples, we obtain a Family of Intervals. Each
member of this Family is a single interval of the form p
± (Z*sdp) computed from a member of the Family of
Samples.
When the Family of Intervals is based on large (n>30)
samples, a fixed percentage of the members of the
Family of Intervals contain the population proportion P.
The actual value of this percentage depends on the
multiplier Z.
Confidence Intervals: Basic
Elements of Interpretation
Briefly identify the population of interest: “The population
consists of …”
Briefly identify the population mean of interest: “We seek
to estimate the population mean …”
Briefly describe the family of samples: Each member of
the Family of Samples is a single random sample of
n=? Members of the population. The FoS consists of
every possible random sample of this type …”
Confidence Intervals: Basic Elements
of Interpretation
Briefly describe the family of intervals (FoS).
“Each member of the FoS yields the following statistics: n
(sample size), m (sample mean) and sd (sample std deviation.
Each member of the FoS yields an interval of the form: [m –
Z*(sd/n), m + Z*(sd/n)]. These intervals collectively form a
Family of Intervals (FoI). Each member of the FoI is an interval
derived from a member of the FoS.”
Apply confidence level to the family of intervals (FoI).
“Approximately ??% of these intervals contain the true population
mean …, and the approximately 100-??% fail.”
Interpret the computed interval.
“If our single interval resides in the ??% supermajority, then the
population mean is contained in the interval, and is …”