Question

If two samples A and B had the same mean and standard deviation, but sample A had a larger sample size, which sample would have the wider 95% confidence interval? Homework Help: Sample B as it has the smaller sample Sample B as its sample is more dispersed Sample A as it has the larger sample Sample A as it comes first

Answer 1

Let the sample means of samples A and B be

The sample standard deviations be

We can estimate the population standard deviation using A and B as

Let

= the sample size of A

= sample size of B

The estimated standard error of mean using A is

The estimated standard error of mean using B is

We have been given that sample A had a larger sample size, that is

That means we can say that the standard error of mean for B is greater than standard error of mean for A

What does the above mean?

It means that if is the sample mean of a randomly selected sample of size from a population

and is the sample mean of a randomly selected sample of size from the population

then we can say that has a larger standard deviation (which is also called the standard error of mean) and hence more dispersed,  compared to

Now we can get the 95% confidence intervals for the true mean of the population as

-- we are using z values assuming that the sample sizes are more than 30, else we would have used a t

the width of 95% confidence intervals for the true mean of the population is

Similarly we can say that the width of 95% confidence intervals for the true mean of the population is

Since we can say that

or we can say that sample B has a wider 95% confidence interval for mean than sample A

It means that we are able to estimate the true value of population mean using sample A more precisely (tighter confidence interval) in comparison to while using sample B, because sample A is larger.