confidence intervals
confidence interval for the mean (CI) = an
interval within which the true mean of the
population is believed to be located
confidence intervals
• statisticians mostly use 95% CI and 99% CI
• 95% covers middle 95% of the distribution
- uses Z = +/- 1.96
confidence intervals
• 99% confidence interval covers middle 99% of
the distribution
- uses Z = +/-2.58
calculating CI with a known SD
CI =
Z(
+ Z(
)
) = margin of error = E
CI =
+ E
2 steps to get CI
1. get the lower end of CI
CI =
-Z(
)
2. get the upper end of CI
CI = + Z (
)
example
A sample of 81 people took an IQ test. The mean
of the IQ for this sample was 102. IQ scores in
the general population are known to have a
standard deviation of 15.
1. Estimate the mean of the population with a
95% confidence interval.
2. Show the CI on a graph.
example
A sample of 81 people took an IQ test. The mean
of the IQ for this sample was 102. IQ scores in
the general population are known to have a
standard deviation of 15.
1. Estimate the mean of the population with a
99% confidence interval.
2. Show the CI on a graph.
confidence, precision
and interval width
A sample of 100 people took an IQ test. The
mean of the IQ for this sample was 102.
At a CI of 95%, find:
a) standard error of the mean
b) the confidence interval
confidence, precision
and interval width
Now calculate the 95% CI for samples of
different sizes with different means.
sample A: n = 81, mean = 99
sample B: n = 100, mean = 103
What is the width of each CI? (the difference
between the upper and lower limits)
calculating CI with an unknown SD
CI =
± t (SE)
SE = estimated standard error
t = t value
estimating the standard error (SE)
SE = s/n
e.g. A survey of a sample of 25 college students
reveals that they spend, on average, $5.50 on
lunch with a standard deviation of s=$1.50.
What is the estimated standard error?
t values and t distributions
• t value = a point along the baseline of a
sampling distribution
• t value depends on size of the sample and
width of the confidence interval
• t distributions are theoretical probability
distributions
- similar to SNC, but shape is determined by
degrees of freedom (df)
t values and t distributions
• degrees of freedom = n-1