c. The best estimate of the unknown population standard deviation,
, is the sample standard deviation s, where:
1 n
x x
s
2
This is obtained from the x
n-1
key on the calculator.
N.B.
n x
x s
2
from x
n
is not used as it underestimates the value of
4. Interval Estimate of Population Parameter Confidence interval
Sometimes it is more useful to quote two limits between which the parameter is expected to lie, together with the probability of it lying in that range. The limits are called the
confidence limits and the interval between them the confidence interval.
The width of the confidence interval depends on three sensible factors: a. the degree of confidence we wish to have in it,
i.e. the probability of it including the truth, e.g. 95; b. the size of the sample, n;
c. the amount of variation among the members of the sample, e.g. for means this the standard deviation, s.
The confidence interval is therefore an interval centred on the point estimate, in this
case either a percentage or a mean, within which we expect the population parameter to lie. The width of the interval is dependent on the confidence we need to have that it does in fact
include the population parameter, the size of the sample, n, and its standard deviation, s, if
estimating means. These last two parameters are used to calculate the standard error, s
n, which is also referred to as the standard deviation of the mean.
The number of standard errors included in the interval is found from statistical tables - either the normal or the t-table. Always use the normal tables for percentages which need
large samples. For means the choice of table depends on the sample size and the population standard deviation:
Population standard deviation
Sample size Known: standard error =
n
Unknown: standard error =
n s
Large Normal tables
Normal tables Small
Normal tables t-tables
5. Interpretation of Confidence intervals
How do we interpret a confidence interval? If 100 similar samples were taken and analysed then, for a 95 confidence interval, we are confident that 95 of the intervals
calculated would include the true population mean. In practice we tend to say that we are 95 confident that our interval includes the true population value. Note that there is only
one true value for the population mean, it is the variation between samples which gives the range of confidence intervals.
6. Confidence Intervals for a Percentage or Proportion
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