18.104.22.168 Confidence limits What we can do is to work out the confidence limits (also known as confidence interval). What they give us is an upper and lower limit within which the population percentage will fall.
In the CASE STUDY example, 30.8% of the sample thought homosexuality should be illegal (Table 22.214.171.124). This might be an overstatement or an understatement. Using probability theory, the details of which we will not go into here (for details, see for example Encyclopaedia Britannica: Probability Theory, accessed 23 October 2016), you can work out what the likely range would be for the population of people in favour of making homosexuality illegal.
This is how you do it. Once again the sums are not difficult (especially if you have a calculator or a computer).
1. Convert the percentage of the sample in favour (30.8%) into a decimal (0.308).
2. Work out what is known as the standard error of the proportion (that is, the standard deviation of the sampling error) using this formula:
standard error = √(p(1 - p))/n
where p = sample proportion (in favour) = 0.308
and n = sample size = 143
Standard error = √(0.308)(0.692)/143 Standard error = √0.213136/143) = √0.00149 = 0.0386
3. Use this to find the confidence limits. The usual ones in social science are the 95% confidence limits which are the limits around the sample percentage that have a 95% chance of containing the population percentage. They are found from:
4. Convert these limits back to percentages and we have a 95% confidence limit of:
23.2% to 38.4% (to one decimal place).
5. This means that there is a 95% probability that the population proportion lies between these two percentages. The population value could therefore be as high as 38.4% or as low as 23.2% given our sample. This does presuppose that the sample is an unbiased random sample of the population.
We can now be more confident than before in relation to hypothesis 1a and say that (assuming a random sample) a majority of the population from which the sample was taken does not support making homosexuality illegal because, at most, only 38.4% are in favour.
Calculate the confidence limits for the population proportion in favour of Clause 28. On the basis of this is there a majority in favour or against the clause?
Once again, note that most statistics packages will work out confidence limits and what is important is that you know what they mean. Samples contain sampling error and confidence limits take account of sampling error. Confidence limits are a range of values, derived from the sample, within which the population value is likely to fall. This allows you to be more confident in making statements about the population on the basis of a sample.
In the same way that a proportion will be subject to sampling error, so will an arithmetic mean. For example, the sample mean age of consent for gays was 22.198 years. The population mean age will similarly lie between confidence limits.
The equivalent formula for 95% confidence limits for a sample mean are:
Upper limit: sample mean + 1.96 (standard error)
Lower limit: sample mean - 1.96 (standard error)
Where the standard error for the mean = s/√(n - 1) and s = sample standard deviation
and n = sample size
So this means that the given the very large variation in the sample, the arithmetic mean age of consent for gays for the population from which the sample was taken could be as low as 19.268 or as high as 25.128.
Calculate the 95% confidence limits for the age of consent for lesbians. Does the result suggest that the average age of consent for lesbians is lower than for gays?