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
The only difference between calculating the interval for percentages or for proportions is that the former total 100 and the latter total 1. This difference is reflected in the formulae
used, otherwise the methods are identical. Percentages are probably the more commonly calculated so in Example 2 we will estimate a population percentage.
The confidence interval for a population percentage or a proportion, , is given by:
n p
100 p
z p
for a percentage or
n p
1 p
z p
for a proportion where:
is the unknown population percentage or proportion being estimated, p is the sample percentage or proportion, i.e. the point estimate for
, z is the appropriate value from the normal tables,
n is the sample size.
The formulae
n p
100 p
and
n p
1 p
represent the standard errors of a percentage
and a proportion respectively. The samples must be large, 30, so that the normal table may be used in the formula.
We therefore estimate the confidence limits as being at z standard errors either side of the sample percentage or proportion. The value of z, from the normal table, depends upon the
degree of confidence, e.g. 95, required. We are prepared to be incorrect in our estimate 5 of the time and confidence intervals are always symmetrical so, in the tables we look
for Q to be 5, two tails.
Example 2
In order to investigate shopping preferences at a supermarket a random sample of 175 shoppers were asked whether they preferred the bread baked in-store or that from the large
national bakeries. 112 of those questioned stated that they preferred the bread baked in- store. Find the 95 confidence interval for the percentage of all the stores customers who
are likely to prefer in-store baked bread. The point estimate for the population percentage,
, is p =
100 175
112
= 64
Use the formula:
n p
100 p
z p
where p = 64 and n = 175 From the short normal table 95 confidence
5, two tails z = 1.96
n p
100 p
z p
175
36 64
96 .
1 64
the confidence limits for the population percentage, , are and
5. Confidence interval for the Population Mean,
