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Confidence Intervals

We are conducting a survey which will provide us with
1 an estimate of the number of companies using a given product
2. The average annual consumption of that product

Each of these estimates will have a confidence interval around it
and I know how to calculate these.

Multiplying the number of companies using the product times the average annual consumption will give me an estimate of the "Total consumption". My question is how do calc a confidence interval around that estimate, give that it is composed of two seperate set of confidence intervals. Any help, greatly appreciated.

What affects confidence intervals?

Recall that two sided confidence intervals are formed as

sample statistic minus margin of error population parameter sample statistic plus margin of error

Population parameter is typically the mean, proportion, or standard deviation

Margin of error is calculated as critical value times

The degree of freedom df is fixed to infinity for the z distribution, but equals sample size minus one, for the t and Chi^2 distributions, so df always increases with sample size n. A larger df with the same level of confidence yields a smaller critical value for the t distribution, but a larger critical vavlue for the Chi^2 distribution.

The buttomline is that, the sample size n with a fixed level of confidence will either leave the critical value unchanged (for the z distribution), or change the critical value (for the other two distributions).

Also the standard error depends on the sample size n (decreases inversely with the square root of n) and increases directly with the sampling variation. The more accurately we know the latter, the smaller the margin of error.

So both the sample size and the sampling variation affect your confidence interval, as does the level of confidence. However your question addresses which has a bigger effect.

The key here is to note that one always builds confidence intervals for unbiased parameters such as mean, proportion, and standard deviation. This means the sample statistic will consistently target the population parameter (as a point estimate).

The conclusion should be that the sample size is what carries more influence in determining the margin of error (affecting both the standard error, and in two out of three cases the critical value). These are ssuming that the level of confidence is fixed, usually 95% or 99% corresponding to an alpha of 5% or 1% respectively.