Transcript Confidence intervals

How confident are we that our
sample means make sense?
Confidence intervals
Maybe not
Confidence intervals
1.
Find a confidence interval for the population mean
using a sample from a normal distribution with
known variance.
2. Find a confidence interval for the population mean
using a sample from any distribution with known or
unknown variance
3. Make inference from confidence intervals.
Point estimate
• A single number that estimates a population
parameter is called a point estimate
How confident we are about an estimate
depends on two factors:
• The size of the sample – the larger the size
of the sample, the closer the estimate is
likely to be to the true population mean
• The variance of the population – if readings
are generally more varied the estimate will be
less reliable

standard error 
From previous lesson
n
• :
where  is the population
standard deviation
When comparing two estimates of the same
or a similar parameters.
Point estimate
Given two samples:
1. Get each of their unbiased estimator of mean (point
estimate)
2. Calculate their standard errors
3. Rule of thumb the one with the smallest standard
error, is more likely to have an estimated mean
nearer to the population true mean.
Task
• Exercise A (page 111)
Difficulty using point estimate
• In complex situations the choice of an estimate of a
population parameter is not always clear. Statisticians
may find that they have no idea how to use the
sample data to estimate the population parameter of
interest, or they may in fact have several equally
plausible competing estimates to select from.
• Point estimates do not inform us about how much the
estimate is likely to be in error.
NORMAL DISTRIBUTION SAMPLE MEANS
Confidence Interval
9.00
6.00
90
 90%
100
Confidence Intervals
• A point estimate is the middle point of the interval
and the endpoints of the interval communicate the
size of the error associated with the estimate and
how ‘confident’ we are that the population parameter
is in the interval
9.00
6.00
Confidence Interval Calculation
• Typical confidence levels used in practice for confidence
intervals are 90%, 95% or 99% with 95% occurring most
frequently
• Find 90% from the
percentage points
table p=0.95 and the value
z=1.6449 ≈ 1.645
Confidence interval=x  1.645

n
Confidence Interval
The higher the level of confidence, the wider
the confidence interval needs to be.
90%
95%
find p=0.95 z=1.6449  1.645
find p=0.975 z=1.96
90% and 95% confidence intervals are given by:

 

, x  1.645
 x  1.645
  90% interval
n
n


 

, x  1.96
 x  1.96
  95% interval
n
n

Task
• Exercise B Page 116
All the questions that you have done so far
have been from populations that have been
normally distributed.
CLT states that if sample sizes are large enough
then the mean of any distribution is approximately
normally distributed with a standard error 
n
therefore, any random sample where n is big enough
will have a 95% confidence interval given by:

 

, x  1.96
 x  1.96

n
n

Using an estimated variance
• When the σ² is not known but n ≥ 30 an
unbiased estimate of the variance S2 can be
2
calculated using
S
2
x  x


i
n 1
x

x  n
2
2

n 1
• So when the variance or standard deviation are
not known replace σ² with S2 and σ with S
Task
• Read examples 4 & 5
• Exercise D page 120
Key points
• A 95% confidence interval tells us that there is a probability of 0.95
that the interval contains the population μ
• If the sample is taken from a normal population then a 95%
confidence interval is given by 



, x  1.96
 x  1.96

n
n


• If the sample is taken from any distribution and n is large enough,
then a 95% confidence interval is given by

 

, x  1.96
 x  1.96

n
n


• If the population is not know then replace σ² with S2 and σ with S
and 95% can by given by:
S
S 

, x  1.96
 x  1.96

n
n

Task
• Mixed questions
– page 122
• Homework or now
– Test yourself
page 122
Confidence intervals
1.
Find a confidence interval for the population mean
using a sample from a normal distribution with
known variance.
2. Find a confidence interval for the population mean
using a sample from any distribution with known or
unknown variance
3. Make inference from confidence intervals.