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,
, when the population standard deviation,
, is known. Example 3
: For the small supermarket as a whole it is known that the standard deviation of the wages for part-time employees is £1.50.
A random sample of 10 employees from the small supermarket gave a mean wage of £4.15 per hour. Assuming the same standard deviation, calculate the 95 confidence interval for
the average hourly wage for employees of the small branch and use it to see whether the
figure could be the same as for the whole chain of supermarkets which has a mean value of £4.50.
As we actually know the population standard deviation we do not need to estimate it from the sample standard deviation. The normal table can therefore be used to find the number
of standard errors in the interval. Confidence Interval:
n z
x
where z comes from the short normal table
10 50
. 1
96 .
1 15
. 4
n z
x
This interval includes the mean, £4.50, for the whole chain so the average hourly wage could be the same for all employees of the small supermarket.
6. Confidence interval for the Population Mean,
, when the population standard deviation is not known, so needs estimating from the sample standard deviation, s.
The population standard deviation is unknown, so the t-table, must be used to compensate for the probable error in estimating its value from the sample standard deviation.
Example 4 : Find the 99 confidence interval for the mean value of all the invoices, in
Example 1, sent out by the small supermarket branch. If the average invoice value for the whole chain is £38.50, is the small supermarket in line with the rest of the branches?
Confidence Interval:
n s
t x
where the value of t comes from the table of percentage points of the t-distribution using n - 1 degrees of freedom
= n - 1 From Example 1
x s
n
£32. ,
£7. , .
92 12
20
degrees of freedom = 20 – 1 = 19;
99 confidence; from tables t = 2.87
n s
t x
= This interval does not include £38.50, so the small branch is out of line with the rest.
7. Comparison of Means using Overlap in Confidence Intervals
